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THE  RAILWAY 
TRANSITION    SPIRAL 


TAL.BOT 


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THE 


Railway  Transition  Spiral 


ARTHUR  N.  TALBOT,  C.  E. 

Member  American  Society  of  Civil  Engineers,  Professor 

of  Municipal  and  Sanitary  Engineering 

University  of  Illinois 


FIFTH   EDITION,    REVISED 

ELEVEXTH    THOUSAND 


NEW  YORK: 
McGRAW-HILL  BOOK  COMPANY 

1915 


COPYRIGHT  1901 
BY 

ARTHUR  N.  TALBOT 


PREFACE 


The  railway  transition  spiral  here  presented  is  a  flex- 
ible easement  curve  of  general  applicability  and  of  com- 
paratively easy  analysis.  The  conceptions  and  methods 
used  are  similar  to  those  of  ordinary  circular  railroad 
curves.  The  definition  is  based  upon  degree-of-curve,  and 
degree-of-curve,  central  angle,  and  deflection  angle  may  be 
calculated,  and  the  curve  may  be  located  by  transit  and 
chain  or  by  co-ordinates  from  tangent  and  circular  curve. 
The  field  work  is  quite  similar  to  that  of  circular  curves. 
The  spiral  is  easily  applied  to  a  variety  of  field  problems 
and  to  a  wide  range  of  location  and  old  track  conditions. 
In  the  principal  formulas,  angles,  co-ordinates,  offsets,  etc., 
are  expressed  in  terms  of  the  length  or  distance  along  the 
spiral.  The  use  of  series  in  the  development  of  the  prop- 
erties permits  an  estimate  of  the  error  involved  in  dis- 
carding negligible  terms 

The  principal  easement  curves  in  use  give  alignments 
which  approach  each  other  very  closely,  so  that,  for  equal 
easements,  it  may  be  expected  that  the  riding  qualities 
will  not  differ  sensibly.  In  general,  ease  of  calculation, 
simplicity  of  field  work,  and  general  applicability  and 
flexibility  will  determine  the  form  of  easement  curve  to 
be  selected.  To  establish  the  underlying  principles  of  an 
easement  curve  of  any  range  requires  considerable  math- 
ematical analysis.  The  ordinary  treatment  of  circular  rail- 
way curves  assumes  previous  knowledge  of  the  geometrical 
properties  of  the  circular  curve,  but  the  properties  of  the 
railway  spiral  must  be  deduced  from  the  beginning.  For- 
tunately the  spiral  is  not  complex,  and  its  properties  prove 
to  be  simple  and  general.  A  general  treatment  of  such  a 
curve  i.as  many  advantages  over  approximate  or  special 
treatments.  Approximate  solutions  may  overlook  important 
variables,  and  short  methods  may  be  limited  to  short  and 
inefficient  easements.  The  range  of  conditions  of  railway 
curves  is  so  wide  that  it  is  best  to  develop  methods  of 
fairly  general  applicability,  and  these  may  then  be  simplified 
in  meeting  individual  conditions. 


380514 


The  treatment  herein  given  has  been  quite  widely  used 
on  the  railroads  of  the  United  States  and  many  engineers 
have  commended  its  simplicity,  convenience,  and  flexi- 
bility. The  methods  and  principles  are  readily  taken  up 
by  instrument  men,  and  the  field  work  has  proved  little 
more  difficult  than  that  for  circular  curves.  The  use  of  a 
regular  rate  of  transition  per  100  feet  of  spiral  is  advan- 
tageous, and  the  tables  are  in  convenient  form. 

The  treatment  of  the  railway  transition  spiral  was 
published  in  Technograph  No.  5,  1890-91,  and  was  pub- 
lished in  field-book  size  in  1899.  Careful  attention  has 
been  given  in  this  revision  to  illustrative  examples  and 
explanations.  The  tables  have  been  extended  and  a  treat- 
ment of  the  Uniform  Chord  Length  Method  and  of  Street 
Railway  Spirals  added.  For  much  of  the  latter,  acknowl- 
edgment is  made  to  Mr.  A.  L.  Grandy.  The  writer  is  indebted 
to  Messrs.  J.  K.  Barker,  Alfred  L.  Kuehn,  and  many  others 
for  valuable  assistance  in  the  preparation  of  tables  and  text. 

URBANA,  ILLINOIS,  A.  N.  T. 

November  11,  1901. 


CONTENTS 


Nomenclature 1 

Use.  Definition.  Notation.  Measurement  of  length. 

Theory -         -         -       6 

Intersection  angle.  Co-ordinate  x  and  y.  Spiral 
deflection  angle.  Table  of  corrections.  Deflection 
angle  at  point  of  spiral.  Ordinates.  Offset.  Ab- 
scissa of  P.  C.  Tangent-distance.  External-dis- 
tance. Long  chord.  Spiral  tangent-distances. 
Middle  ordinate. 

Summary  of  Principles       -  17 

Principal  formulas.  Angles  and  deflection  angles. 
Angles  from  tangent.  Angles  from  chord.  Diverg- 
ence from  osculating  circle.  Co-ordinates.  Offset. 
Other  distances. 

Description  and  Use  of  the  Tables  -  -     24 

Tables  for  transition  spiral.  Accuracy.  Interpola- 
tion. General  use  of  Table  IV.  Corrections  for 
calculations.  Other  tables. 

Choice  of  a  and  Length  of  Spiral      -        -        -        -        -     28 
Effect   of    speed.      Attainment    of    superelevation. 
Amount  of  superelevation.     Minimum  spiral.     Se- 
lection of  spiral. 

Location  of  P.  S.,  P.  C.  C.,  and  P.  C.  -         -         -         -         -     33 

Laying  out  the  Spiral  by  Co-ordinates     -        -        -        -     35 
From  tangent,  tangent  and  curve,  and  spiral  tan- 
gents. 

Location  by  Transit  and  Deflection  Angles      -        -        -     38 
Transit  at  P.  S.    Transit  on  spiral.    Intermediate  de- 
flection angles.    Transit  at  P.  C.  C.    Transit  notes. 


Application    to    Existing    Curves 44 

To  replace  the  entire  curve — Two  methods.  To  re- 
place part  of  the  curve.  To  re-align  and  com- 
pound. Methods  of  trackmen. 

Compound   Curves -     54 

General  method.     To  insert  in  old  track. 

Miscellaneous   Problems -     60 

To    change    tangent   between    curves    of    opposite 
direction.      To    change    tangent   between    curves    of 
same  direction. 

Uniform   Chord    Length    Method       -  -     64 

Formulas.  Tables.  Fractional  chord  lengths. 
Use  of  method. 

Street  Railway  Spirals       -  72 

Theory.  Tables.  Laying  out.  Arc  excess.  Curv- 
ing rails.  Double  track. 

Conclusion 79 

Explanation   of  Tables 84 

Tables  I-XI.  Transition  Spirals  -  -  -  86-97 
Table  XII.  Factors  for  Ordinates  97 

Table  XIII.     Unit  Spiral  Deflection  Angles      -  98 

Table  XIV.  Coefficients  for  Deflection  Angles  -  -  99 
Tables  XV-XIX.  Street  Railway  Spirals  -  -  100-101 
Table  XX.  Offsets  for  Spirals 102 


Railway  Transition  Spiral 


NOMENCLATURE 

1 .  A  transition  curve,  or  easement  curve,  as  it  is  some- 
times called,  is  a  curve  of  varying  radius  used  to  connect 
circular  curves  with  tangents  for  the  purpose  of  avoiding 
the  shock  and  disagreeable  lurch  of  trains  due  to  an  in- 
stant change  in  the  relative  position  of  cars,  trucks,  and 
draw-bars  and  also  to  a  sudden  change  from  level  to 
inclined  track.  With  the  spiral  the  superelevation  of  the 
outer  rail  may  be  made  to  correspond  to  the  curvature 
at  all  points  around  the  transition  curve,  and  the  trucks, 
springs,  draw-bars,  and  car  body  will  gradually  attain  their 
final  position  for  the  main  curve.  The  primary  object 
of  the  transition  curve,  then,  is  to  effect  smooth  riding 
when  the  train  is  entering  or  leaving  a  curve. 

The  generally  accepted  requirement  for  a  proper  tran- 
sition curve  is  that  the  degree-of-cruve  shall  increase 
gradually  and  uniformly  from  the  point  of  tangent  until 
the  degree  of  the  main  curve  is  reached,  allowing  the 
superelevatic' ;  to  increase  uniformly  from  zero  at  the 
tangent  to  tVie  full  amount  at  the  connection  with  the 
main  curve  and  yet  to  have  at  every  point  the  appropriate 
superelevation  for  the  curvature.  In  addition  to  this,  an 
acceptable  transition  curve  must  be  so  simple  that  the 
field  work  may  be  easily  and  rapidly  done,  and  should  be 
so  flexible  that  it  may  be  adjusted  to  meet  the  varied 
requirements  of  problems  in  location  and  construction. 

No  attempt  will  here  be  made  to  show  the  necessity  or 


the  utility  of  transition  curves.  The  principles  and  some 
of  the  applications  of  one  of  the  best  of  these  curves, 
the  railway  transition  spiral,  will  be  considered. 

2.  Definition.     The  Transition  Spiral  is  a  curve  whose 
degree-of-curve  increases  directly  as  the  distance  along 
the  curve  from  the  point  of  spiral. 

Thus,  if  the  spiral  is  to  change  at  the  rate  of  10°  per 
100  feet,  at  10  feet  from  the  beginning  of  the  spiral  the 
curvature  will  be  the  same  as  that  of  a  1°  curve;  at  25 
feet,  as  of  a  2°30'  curve;  at  60  feet,  as  of  a  6°  curve. 
Likewise,  at  60  feet,  the  spiral  may  be  compounded  with 
a  6°  curve;  at  80  feet,  with  an  8°  curve,  etc. 

This  curve  fulfills  the  requirements  for  a  transition 
curve.  Its  curvature  increases  as  the  distance  measured 
around  the  curve.  The  formulas  for  its  use  are  compar- 
atively simple  and  easy.  The  field  work  and  the  com- 
putations necessary  in  laying  it  out  and  in  connecting  it 
with  circular  curves  are  neither  long  nor  complicated, 
ar?H  are  similar  to  those  for  simple  circular  curves.  The 
curve  is  extremely  flexible,  and  may  easily  be  adapted  to 
the  requirements  of  varied  problems.  The  rate  of  change 
of  degree-of-curve  may  be  made  any  desirable  amount 
according  to  the  curve  used,  the  maximum  speed  of  trains, 
or  the  requirements  of  the  ground. 

As  the  derivation  of  the  formulas  is  somewhat  long, 
their  demonstration  will  be  given  first.  The  explanation 
and  application  of  these  formulas  to  the  field  work  and 
to  the  computations  will  be  given  separately,  a  knowledge 
of  the  demonstration  not  being  essential  to  the  application. 

3.  In  Fig.  1,  DLH  is  the  circular  curve  and  AP  the 
prolongation  of  the  initial  tangent  wh:ch  are  to  be  con- 
nected by  the  transition  spiral.     D  is  the  point  where  the 
completed  circular  curve  gives  a  tangent  DN  parallel  to 
the  tangent  AP,  and  will  be  called  the  P.  C.  of  the  cir- 
cular curve.    AEL  is  the  transition  spiral  connecting  the 


NOMENCLATURE  3 

initial  tangent  AP  with  the  main  or  circular  curve  LH. 
A  is  the  beginning  of  the  spiral  and  will  be  known  as 
P.S.,  point  of  spiral.  L  is  the  beginning  of  the  circular 
curve  LH,  and  will  be  called  P.C.C.,  point  of  circular 
curve.  AP  will  be  used  as  the  axis  of  X,  and  A  as  the 
origin  of  co-ordinates;.  BD  is  the  offset  between  the 
tangent  AB  of  a  circular  curve  and  spiral,  and  the  parallel 
tangent  DN  of  an  unspiraled  curve. 

The  degree-of-curve  of  the  spiral  at  any  point  is  the 
same  as  the  degree  of  a  simple  curve  having  the  same 
radius  of  curvature  as  the  spiral  has  at  that  point.  The 
radius  of  the  spiral  changes  from  infinity  at  the  P.S. 
to  that  of  the  main  curve  at  the  P.C.C.  The  spiral  and 
a  simple  curve  of  the  same  degree  will  be  tangent  to 
each  other  at  any  given  point;  i.  e.,  they  will  have  a 
common  tangent. 

4.     The  following  notation  will  be  used: 

P.S.  =  Point  of  Spiral.   (A,  Fig.  1.) 

P.C.C.  =  point  where  spiral  compounds  with  circular, 
curve.     (L,  Fig.  1.) 

P.C.=beginning  of  offsetted  circular  curve.     (D,  Fig.  1.) 

R  =  radius  of  curvature  of  the  spiral  at  any  point. 

D  =  Degree-of-curve  of  the  spiral  at  any  point;  some- 
times called  D±  at  the  end  of  spiral.  Generally  D^  is 
made  the  same  as  DQ,  the  degree  of  the  main  curve. 

a  =  rate  of  change  of  the  degree-of-curve  of  the  spiral 
per  100  ft.  of  the  length.  It  is  equal  to  the  degree-of- 
curve  of  the  spiral  at  100  ft.  from  the  P.S. 

j  =  length  in  feet  from  the  P.S.  along  the  curve  to  any 
point  on  the  spiral. 

L  =  number  of  100- ft.  stations  from  the  P.S.  along  the 
curve  to  any  point  on  the  spiral;  in  other  words  the  dis- 
tance to  any  point  measured  in  units  (or  stations)  of  100 
ft.  For  the  whole  spiral  (to  P.C.C.)  it  is  sometimes 
called  Lt. 


4-  NOMENCLATURE 

/=  Total  central  angle  of  the  whole  curve  (intersec- 
tion angle),  or  twice  BCH  of  Fig.  1,  H  being  the  middle 
of  the  circular  arc. 

A  =  angle  showing  the  change  of  direction  of  the  spiral 
at  any  point,  and  is  the  angle  between  the  initial  tangent 
and  the  tangent  to  the  spiral  at  the  given  point.  For 
the  whole  spiral  it  is  equal  to  PTL  and  may  be  called  Ax. 
The  lastter  is  also  equal  to  DCL. 

0  =  spiral  deflection  angle  at  the  P.S.  from  the  initial 
tangent  to  any  point  on  the  'Spiral.  For  the  point  L  (Fig. 
1)  it  is  BAL. 

<$  —  deflection  angle  at  any  point  on  the  spiral,  between 
the  tangent  at  that  point  and  a  chord  to  any  other  point. 
At  L,  for  the  point  A,  <1>  is  TLA. 

x  =  abscissa  of  any  point  en  the  spiral,  referred  to  the 
P.S.  as  the  origin  and  the  initial  tangent  as  the  axis 
of  X.  For  the  point  L,  .ar  =  AM. 

3/z^ordinate  of  the  same  point,  measured  at  right  angles 
to  the  above  axis.  For  the  point  L,  y  =  ML. 

t  =  abscissa  of  the  P.C.  of  the  main  curve  produced 
backward;  i.e.,  of  a  simple  curve  without  the  spiral. 
For  P.C.  at  D,  t  =  AB. 

o  =  offset  between  the  initial  tangent  and  the  parallel 
tangent  from  the  main  curve  produced  backward,  or  it 
is  the  ordinate  of  the  P.C.  of  the  produced  main  curve. 
If  D  is  the  P.C.,  BD  is  o.  It  is  also  the  radial  distance 
between  the  concentric  circles  LH  and  BK. 

T  =  tangent-distance  for  spiral  and  main  curve  =  dis- 
tance from  A  to  the  intersection  of  tangents. 

E  =  external-distance  for  spiral  and  main  curve. 

C  =  long  chord  AL  of  the  transition  spiral. 

u  =  distance  along  initial  tangent  from  P.S.  to  inter- 
section with  spiral  tangent  =  AT  for  point  L. 

v  =  length  of  spiral  tangent  to  intersection  with  initial 
tangent  =  TL  for  point  L. 


NOMENCLATURE  5 

5.  The  length  of  the  spiral  is  to  be  measured  along 
chords  around  the  curve  in  the  same  way  that  simple 
curves  are  usually  measured,  using  any  length  of  chord 
up  to  a  limit  which  depends  upon  the  degree-of-curve  of 
the  spiral.  The  best  railroad  practice,  in  the  writer's 
opinion,  considers  circular  curves  up  to  a  7°  curve  as 
measured  with  100-ft.  chords,  from  7°  to  14°  as  measured 
with  50-ft.  chords,  and  from  14°  upwards  as  measured 
with  25-ft.  chords;  that  is  to  say,  a  7°  curve  is  one  in 
which  two  50-ft.  chords  together  subtend  7°  of  central 
angle,  a  14°  curve  one  in  which  four  25-ft.  chords  to- 
gether subtend  14°  of  central  angle.  The  advantages 


FIG.  1 

of  this  method  are  two-fold, — the  length  of  the  curve 
as  measured  along  the  chords^  more  nearly  approximates 
the  actual  length  of  the  curve,  and  the  radius  of  the 
curve  is  almost  exactly  inversely  proportional  to  the 
degree-of-curve.  The  latter  consideration  is  an  important 
one,  simplifying  many  formulas.  With  this  definition  of 
degree-of-curve,  the  formula  R  =  5^°  will  give  no  error 

greater  than  1  in  2,500.  For  a  10°  curve  the  error  in 
the  radius  is  .15  feet,  and  for  a  16°  curve  .06  feet.  This 
approximate  value  of  R  will  give  a  resulting  error  in 
the  length  of  the  spiral,  for  the  ordinary  limits  of  spirals, 


t>  THEORY 

of  less  than  -^Q-Q  of  the  length,  and  will  not  reach  0,1  ft. 
The  resulting  error  in  alignment  is  -oVo^  and  will  not 
reach  0.01  ft.  The  difference  between  the  length  of  the 
curve  and  that  of  these  chords  is  less  than  1  in  7000. 
For  spirals  measured  with  lengths  of  chords  as  here 
specified,  or  shorter,  the  error  either  in  alignment  or  dis- 
tance will  be  well(within  the  limits  of  accuracy  of  the  field 
work,  and  hence  the  relation  R  =  5^°  will  be  considered 
true. 


THEORY 

6.  Intersection  angle  A. — From  the  definition  of  the 
transition  spiral,  we  have,  remembering  that  the  value  of 
a  as  defined  above  requires  the  length  of  curve  to  be 
measured  in  100-ft.  units  (stations)  instead  of  feet, 


For  the  P.C.C.  this  becomes  D1==aLr 
From  the  calculus  the  radius  of  curvature 

R  =  —  . 

Substituting  the  expression  R  =    ^    and  solving, 
as  ds 


573000  * 

as2  a  L2 

Integrating,  A  =  - 


1146000       114.6  ' 

Changing  A  from  circular  measure  to  degrees, 
A  =  JaL2  ....................................  (2) 

which  is  the  intrinsic  equation  of  the  Transition  Spiral. 
For  the  P.C.C.  this  becomes  ^  =  J  a  L*. 


Since  from  (i)  a  =  -=—,  we  also  have 


(3) 


CO-ORDINATES  7 

From  these  equations  it  will  be  seen  that 

(a)  the  change  of  direction  of  the  spiral  varies  as  the 
square  of  the  length  instead  of  as  the  first  power  of  the 
length  as  in  the  simple  circular  curve,  and 

(b)  the  transition  spiral  for  any  angle  A  will  be  twice 
as  long  as  a  simple  circular  curve. 

7.  Co-ordinates  a?  and  y.  To  find  the  co-ordinates, 
x  and  y,  of  any  point  on  the  spiral,  we  have  by  the  ca'i- 
culus  dy=ds  sin  A  and  dx=ds  cos  A.  Expanding  the  sine 
and  cosine  into  infinite  series,  substituting  for  ds  its  value 
in  terms  of  dA,  and  integrating,  we  have 


X  = 


As  A  here  is  measured  in  circular  measure  and  is  only 
J  when  the  angle  is  28°. 65,  these  series  are  rapidly  con- 
verging, especially  for  smaller  angles. 

Changing  the  angle  A  from  circular  measure  to  degrees, 
substituting  for  A  the  value  given  in  (2),  and  dropping 
the  small  terms, 

3;  =  .291  a  Is— -.00000158  a3L7.. (6) 

For  values  of  A  less  than  15°  the  last  term  may  be 
dropped,  and  up  to  25°  the  term  will  be  small.  D  U  may 
also  be  written  in  place  of  a  L3.  For  all  except  extreme 
lengths,  the  last  term  may  be  dropped.  Using  y  =  291aLs, 
it  is  seen  that  y  varies  as  the  cube  of  the  distance  of  the 
point  from  the  P.S. 

Likewise  changing  A  from  circular,  measnre  to  degrees 
etc.,  equation  (5)  becomes 

x  =  100  L  —  .000762  a2  L5  +  .0000000027   a4  L9 (7) 

Or  x  =  100  L  —  .000762  D2  U (8) 


8  THEORY 

The  second  term  in  second  member  of  equation  (7)  or 
(8)  may  be  used  as  a  correction  to  be  subtracted  from 
the  length  of  the  curve  in  feet.  The  last  term  in  equation 
(7)  can  be  omitted,  except  for  extreme  lengths. 

8.  Spiral  deflection  angle  0.  —  It  is  desired  to  find 
the  deflection  angle  0  for  any  point  on  the  spiral,  as  BAL 
for  the  point  L  (Fig.  1).  To  show  that  this  is  nearly 
JA,  divide  equation  (4)  by  equation  (5). 

tan  9  =  -J-  A  +  ^  A*  +  Tirf  ^-5-  J5,  etc.  But  from  the 
tangent  series  for  JA, 

tan  i-  A  =  i  A  +  -Jr  J3  +  ^3-  J5,  etc.  Subtracting  one 
from  the  other,  we  get  a  series  which  is  rapidly  decreas- 
ing when  A  is  less  than  40°.  Investigating  this  differ- 
ence, remembering  that  A  is  in  circular  measure,  it  is 
found  that  the  error  of  calling  the  two  equations  equal 
is  less  than  1'  for  A  =  25°  and  decrease  1  rapidly  below 
this.  As  A  will  rarely  reach  25°,  and  aL  \he  discrepancy 
is  only  a  small  fraction  of  a  minute  for  any  angle  ordi- 
narily used,  and  as  the  resultant  error  of  direction  will 
be  corrected  at  the  P.C.C.  when  A  —  0  is  turned  off,  we 
may  ordinarily  disregard  this  and  write 

* 

(9) 


CL 

where  0  is  in  degrees. 

From  equation  (9)  it  is  seen  that  the  spiral  deflection 
angles  to  two  points  on  the  spiral  will  be  to  each  other 
as  the  square  of  the  distances  to  the  points. 

9.  The  error  in  equation  (9)  is  dependent  upon  the 
value  of  A  or  0  and  hence  may  be  expressed  independently 
of  the  length  of  spiral  and  rate  of  transition.  For  A 
between  20°  and  40°,  the  number  of  minutes  correction 
to  be  subtracted  from  J  A  or  J  a  L2  to  give  0  is  .000053  A3 
where  A  is  in  degrees.  The  following  table  gives  the 
deductions  for  various  angles,  and  for  other  values  inter- 
polations may  be  made: 


SPIRAL  TANGENT  9 

Correction   in   minutes   to  be   subtracted  from  J   A   or 
a  L2  to  give  more  precise  values  of  6. 


A 

12° 
15° 

18° 


Cor. 
0.1 
0.2 
0.3 


A 

21° 

24° 
27° 


Cor. 
0.5 
0.7 
1.0 


A 

30° 
33° 
36° 


Cor. 
1.4 
1.9 

2.4 


Thus  when  A  is  18°,  the  real  value  of  0  will  be  6°— 
0'.3  =  5°  59'.7.  For  a  value  of  0  near  6°  the  same  correc- 
tion may  be  made.  It  will  be  seen  that  for  the  spiral  de- 


W      A 


FIG.  2 


flection    angles    ordinarily    used    the    correction    may    be 
neglected  without  material  error. 

For  the  terminal-point  of  the  spiral,  the  P.C.C.,  the 
value  of  0j  may  be  obtained  from  equation  (9).  In  the 
extreme  cases,  where  a  further  term  is  needed,  the  cor- 
rection may  easily  be  made  from  the  above  table. 

10.  Spiral  tangent.  To  find  the  tangent  at  any  point 
of  the  spiral,  L,  lay  off  a  deflection  angle  from  LA  equal 
to  A — 0.  When  A  is  not  over  20°,  §  A,  or  20  may  be 
used.  This  since  FLT  =  PTL  =  A,  and  FLA  =  PAL  =  0. 
This  is  true  for  any  point. 


10  THEORY 

For  the  terminal  point  of  the  spiral,  P.C.C.,  this  be* 
comes  A±  —  B1  which  is  generally  expressed  with  sufficient 
precision  by  §  Ar 

11.     Deflection    angle  at  point   on   spiral.  —  The  de- 

flection angle  from  the  tangent  at  any  point  on  the  spiral 
to  locate  a  second  point  may  be  found  as  follows  :  In  Fig. 
2,  let  U  be  the  distance  from  the  P.S.  to  R,  and  L  the 
distance  from  the  P.S.  to  any  other  point  on  the  spiral, 
as  K.  Let  FRN  be  the  tangent  at  R,  and  RFM  =  A'  its 
angle  with  the  initial  tangent,  and  ff  the  corresponding 
spiral  deflection  angle,  RAF.  Let  KTM=A  be  angle  of 
tangent  at  K  with  initial  tangent,  equal  to  total  change 
of  direction  of  the  spiral  up  to  that  point,  ff  and  9  are 
the  deflection  angles  at  the  P.S.  for  R  and  K  respectively. 
=  <£=  required  deflection  angle. 


To  show  that  angle  <1>+A'  is  almost  exactly  the  same 

3.          Jl 
as  the  angle  i^4  I  -  r"0r  £  (J+J*J'*+  A'),  the  fol- 

- 


_ 

lowing  somewhat  long  and  tedious  operation  may  be  gone 
through.  It  is  thought  not  necessary  to  give  it  in  detail 
here. 


X  —  X\ 

Substitute  for  the  co-ordinates  in  the  above  equation  their 
values  from  equations  (4)  and  (5),  and  also  develop 

fo 
—  J'"* 
tan  i  -^  —    —  -  into  a  series,  and  subtract  the  latter 

A^-  A'% 

from  the  former.  An  expression  for  the  difference  will  be 
found,  which  amounts  to  but  a  small  fraction  of  a  minute 
for  any  value  of  A  up  to  35°.  Hence  we  may  write 


DEFLECTION   ANGLE  11 

By  substitutng  for  A  and  A'  their  values  in  terms  of  L 
and  L'  and  reducing,  the  following  value  for  <£  is  found: 


(L  —  L')zhja    (L  —  L')2  ........  ...(10) 

Also,  A'  —  P±3>  =  0  +  \D'L  .....  .......  .  ........  (11) 

And  &'±&  =  ff  +  6  +  \D'L  ..............  ,.'.  ,  ......  (12) 

Even  for  very  large  angles  these  equations  are  quite 
accurate  if  the  exact  value  of  0  is  used.  In  equation  (10) 
the  last  term  ^d(L  —  L')2  should  receive  the  same  cor- 
rection as  an  equal  value  of  0.  Of  course,  for  any  angle 
ordinarily  used  no  correction  need  be  made.  See  cor- 
rection for  0,  page  9. 

12.  In  equation   (10)   it  will  be  noticed  that  the  first 
term   (J  a  L'   (L  —  L')  )   is  equal  to  the  deflection  angle 
for  a  simple  circular  curve  of  the  same  degree  as  the 
spiral  at  the  point  R   (i.e.,  a  I/)   and  of  a  length  equal 
to  the  distance  between  the  two  points  ;  while  the  second 
term  (%a(L  —  L)2)  is  equal  to  the  spiral  deflection  angle 
at  the  P.S.  from  the  initial  tangent  for  an  equal  length  of 
spiral  (L—U). 

If  the  point  to  be  located  had  been  chosen  on  the  side 
of  R  nearer  to  the  P.S.,  the  two  terms  of  equation  (10) 
would  have  opposite  algebraic  signs,  and  the  difference 
of  the  two  quantities  would  be  used.  To  show  that  the 
arithmetical  difference  of  the  two  terms  is  to  be  used  for 
a  point  nearer  the  P.S.  when  the  distance  (L  —  L')  is 
used  without  regard  for  the  algebraic  sign,  equation  (10) 
has  been  written  with  the  plus  and  minus  sign. 

13.  The  spiral  then  deflects  from  a  circle  of  the  same 
degree-of-curve  at  the  same  rate  that  the  spiral  deflects 
from  the  initial  tangent  at  the  beginning.     D'RH,  in  Fig. 
2,   represents  the  circular  curve  tangent  to  spiral  at  R, 
the  two  having  the  same  radius  at  that  point  and  both 


12  THEORY 

being  tangent  to  FRN.  The  deflection  angles  between 
points  on  the  spiral  and  on  the  circle  RH,  and  also  be- 
tween the  spiral  and  RD'  are  the  same  as  the  spiral 
deflection  angle  for  an  equal  length  of  spiral  from  A.  In 
•  the  same  way  at  K,  RKT  =  SKT  —  SKR,  the  latter  angle 
being  equal  to  the  deflection  from  initial  tangent  at  A 
for  a  length  of  spiral  equal  to  KR. 

14.  Equation  (11)   shows  that  the  angle  at  any  point 
between  the  chord  joining  this  point  with  the  P.S.  and  a 
chord  to  any  other  point  (the  angle,  Fig.  2,  between  AR 
produced  and  RK  if  the  point  K  is  to  be  located  from  R) 
is  equal  to  the  spiral  deflection  angle  at  the  P.S.,  0,  for 
the  point  to  be  located    (KAM)    plus  one  third  of  the 
deflection  angle  for  a  circular  curve  of  the  same  degree 
as  that  of  the  spiral  at  the  vertex  of  the  angle,  R,  and  of 
the  length  of  the  spiral  from  P.S.  to  the  point  K.     This 
is  true  whether  the  point  to  be  located  is  nearer  the  P.S., 
or  farther,  than  the  point  used  as  the  vertex  of  the  angle. 

It  may  also  be  readily  shown  from  (2)  that  the  dif- 
ference in  direction  of  the  two  tangents,  A — A',  is  the 
central  angle  for  this  simple  curve  plus  the  spiral  angle, 
both  for  a  length  equal  to  the  distance  between  the  two 
points. 

15.  Equation    (12)   gives  the  value  of  the  deflection 
angle  from  a  line  parallel  to  the  initial  tangent,  the  spiral 
deflection   angle  ff   for  the  point  R  being  added  to  the 
values  in  equation  (11). 

16.  Ordinates   from  osculating   circle.    It  may  also 
be  shown  that  the  offset  distance  between  a  point  on  the 
spiral  and  one  on  the  osculating  circle  is  the  same  as  the 
ordinate  y  from  the  initial  tangent  at  a  point  the  same 
distance  from  the  P.S.  as  the  former  point  is  from  the 
point  of  osculation.     These  ordinates  may  be  measured  is, 
a  direction  normal  to  the  circular  curve. 


OFFSET  13 

17.  Offset  o.— From  Fig.  1,  BD  =  BF— DF  =  BF— 
CDversDCL.  But  o  =  BD,  BF  =  ;y  for  the  end  of  spiral, 
DCL  =  A  foi  the  whole  spiral,  and  CD  =  R.  Hence,  o  = 
y — R  vers  A,  Substituting  for  y,  R  and  A  their  values  in 
terms  of  the  length  of  the  whole  spiral,  applying  the 
versed  sine  series,  and  reducing  we  have  for  o  in  feet 

o  =  .0727  a  L*  =  .0727  £>,  L*. (13) 

where  I>1  and  Li  refer  to  the  whole  length  of  the  spiral. 
The  other  terms  of  the  series  are  so  small  that  they  may 
bo  dropped  when  A  is  less  than  30°.  The  next  term  is 
— .0000002  a3.L7.  It  will  be  seen  that  o  is  approximately 
one  fourth  of  the  ordinate  of  the  P.C.C.,  which,  of  course, 
should  be  true  if  E,  the  middle  point  of  the  spiral,  is 
opposite  D,  the  P.C. 

18.     Offset    given.     From  (13)  and  (3)  we  have 
L  =  3.709  J-^-    (14) 


A  =  1.857 i/  o  D (15) 

y3 

- (16) 


3f,  If  and  -jV  ma7  be  used  for  these  co-efficients  with 
advantage. 


19.     Abscissa  of  P.C.  ,  t.—  From  Fig.  1,  J  =  AB  = 
—  BM  =  ,r  —  FL  =  jr  —  R  sin  A.     Expanding  and  reducing, 

1  =  50  L,—  .000127  a2  L*     ) 

or  [  ...................  (17) 

t  =  50  L,—  .000127  D,2  L,3  ) 

It  should  be  noted  that  the  full  length  of  the  spiral  is 
used  in  the  formula.  The  last  term  may  be  used  as  a 
correction  to  be  subtracted  from  the  half  length  of  the 
spiral.  It  is  easily  tabulated  for  the  principal  spirals, 
and  corrections  for  other  spirals  may  be  found  by  multi- 


14  THEORY 

plying-  the  value  with  a  =  1  for  the  given  length  of  spiral 
by  the  square  of  the  a  used. 

20.  A   comparison   of   t  with   the   abscissa   found  by 
substituting  J  L±  in  equation  (8)  shows  that  BD  cuts  the 
spiral  at  a  point  only  .0001  a3  L8  feet  from  the  middle 
point  of  the  spiral.     This  is  f  of  the  correction  used  in 
equation  (1Y)   for  finding  t  from  J  Lr     For  our  purpose 
we  may  say  that  BD  bisects  the  spiral.     It  also  follows 
that  the  spiral  bisects  the  line  BD,  since  BE  =  £  y.     This 
is  subject  to  slight  error  for  large  angles. 

The  length  of  the  spiral  from  the  P.S.  to  BD,  therefore, 
exceeds  t  by  one  fifth  of  the  t  correction,  and  the  remain- 
der of  the  spiral  exceeds  the  length  of  the  circular  curve 
from  the  P.C.  to  the  P.C.C.  by  four  fifths  of  the  t  correc- 
tion. The  entire  length  of  the  spiral  exceeds  the  distance 
measured  on  t  (AB,  Fig.  1)  plus  the  distance  measured 
around  the  circular  curve  (DL,  Fig.  1)  by  the  t  correc- 
tion given  in  equation  (IT). 

2 1 .  Tangent-distance    T. — To  find  T,  consider  in  Fig. 
1  that  AB  intersects  CH,  H  being  the  middle  of  the  cir- 
cular curve,  at  some  point  P  outside  the  diagram.     Then 
T  =  AP  =  AB  +  BP.     BP  =  BC  tan  BCH. 

Hence  T  =  t+  (R  +  o)  tan  J  / (18) 

t  and  o  tan  -J  /  may  be  computed  separately  and  added 
to  the  T  found  from  an  ordinary  table  of  tangent-dis- 
tances. 

22.  Equation      (18)   gives  T  for  the  same  transition 
spiral  at  each  end  of  the  main  curve.    It  may  be  desirable 
to  make  one  spiral  different  from  the  other.     To  find  an 
expression  for  the  tangent-distance  for  this  case  proceed 
as  follows :     In  Fig.  3,  let  RS  =  HD  =  02,  BD  ==  olt  AB 
=  tv   RT  =  f2,   AE  =  T1,   TE  =  T2,   R  =  radius   of  main 
curve  DLKS,  R  +  02  =  radius  of  HR,  and  /  =  angle  PER. 


TANGENT  DISTANCE  15 

Then  7\  =  f,  +  HC  —  PE,  and 

Ti  =  t1+(R  +  os)  tan  i/  —  (o  —  os)  cot/ (19) 

Similarly,  Ts  =  ta+ (R  +  os)  tan  J/  +  (^  —  02)   cosec  /. 
When  /  is  more  than  90°,  the  last  term  of  (19)  becomes 
essentially  positive. 


23.     External-distance     E—  In  Fig.  1,  E  =  HP.     HP 
=  KP  +  HK.     Hence 


E=(R 


exsec 


(20) 


24.  Long  chord  C.—  In  Fig.  1,  C  =  AL.  ML  =  AL  sin 


MAL,  or  C=- 


.   Putting  this  in  terms  of  the  length 


sin  6' 
of  the  curve, 

C  =  100  L  —  (.000338  a2L5  or  .000338  £>2L3 (21) 

in  which  C  is  in  feet  and  L  in  stations.  It  will  be  seen 
that  the  last  or  correction  term  is  four  ninths  of  the  cor- 
rection for  x  as  given  in  equation  (7).  When  the  cor- 


16  THEORY 

rection  term  for  x  is  tabulated  or  otherwise  known,  the 
length  of  the  long  chord  may  conveniently  be  calculated 
by  subtracting  four  ninths  of  this  x  correction  term  from 
the  whole  length  of  the  spiral. 

The  length  of  a  chord  which  does  not  go  through  the 
P.S.  may  be  calculated  from  the  triangle  formed  by  it 
and  the  two  long  chords  drawn  from  its  ends  to  the  P.S. 
For  all  except  extreme  lengths,  a  chord  may  be  taken  as 
having  the  same  length  as  the  chord  of  an  equal  circular 
arc  whose  radius  is  the  same  as  that  at  the  middle  point 
of  the  given  spiral  arc. 

25.  Spiral  tangent-distances  In.  Fig.  1,  w=,AT  = 
AM  —  MT.  As  MT  —  y  cot  A  or  v  cos  A, 


u  =  x  —  y  cot  A. 
u  =  x  —  v  cos  A . 


}(22) 


Also  v  =  TL  =-^-  —    Expanding  sin  A  into  series,  sub- 
sin  A. 

stituting  the  value  of  y  from  equation  (6),  and  reducing, 


v=    ^      —          L  +  .  000244  a2L5..  ..(23) 

sin  A          3 

The  last  term  is  almost  exactly  one  third  the  correspond- 
ing term  in  equation  (7),  and  hence  v  may  be  found  by 
adding  one  third  of  the  correction  term  used  for  deter- 
mining x  to  one  third  of  the  length  of  the  spiral  in  feet. 

26.  Middle  ordinate.  The  middle  ordinate  for  any 
arc  of  the  spiral  is  equal  to  the  middle  ordinate  for  an 
equal  length  of  circular  curve  of  the  same  degree-of- 
curve  as  the  spiral  at  the  middle  point  of  the  arc  consid- 
ered. This  degree-of-curve  is  the  mean  of  the  D's  at 
the  end  of  the  given  arc.  This  is  an  approximate  formula 
which  is  true  whether  one  end  of  the  chord  is  at  the  P.S. 
or  not. 


SUMMARY    OF    PRINCIPLES  17 

The  ordinate  from  any  other  point  along  a  chord  may 
be  found  as  follows:  Since  the  spiral  diverges  from  the 
osculating  circle  at  the  middle  point  of  the  arc  at  the 
same  rate  as  from  the  initial  tangent,  the  amount  of  this 
divergence  may  be  calculated  by  the  method  given  on 
page  14  and  added  to  or  subtracted  from  the  ordinate  for 
the  osculating  circular  curve.  For  a  point  nearer  the 
P.S.  than  the  center  of  the  spiral  arc,  the  divergence  will 
be  added  to  the  ordinate  of  the  circular  arc,  and  for  one 
farther  away  it  will  be  subtracted  from  the  ordinate.  As 
before,  the  degree  of  the  osculating  circular  curve  is 
the  mean  of  the  D's  at  the  end  of  the  given  arc. 

Other  properties  may  be  found  by  ordinary  trigono- 
metric operations. 


SUMMARY  OF  PRINCIPLES 

27.     For  convenience   of   reference   the   principal   for- 
mulas will  be  repeated  here. 

D  =  aL  and  L  =  —   .......................    I 

CD 
Dl  =  aL1  for  whole  spiral  ................  ...  j 


(2) 
A  =  J^/  =  J-^^i  for  whole  spiral  ...........  j 

y  =  .291  aL3—  etc  ...........................    (6) 

jtr  =  100  L  —  .000762  a2  L5+  etc  ................    (7) 


"  \  (9) 

0±  =  JAj.  =  J0-^i2  for  whole  spiral . 


18  SUMMARY   OF    PRINCIPLES 

3>  =  iaL'   (L  —  L')=±=ja    (L  —  L')2  ...........  (10) 

A'  —  ^=t:^  =  r 

(12) 
=  .0727  aL±3  =  .0727  I\  L^1  ..................  (13) 


L1  =  3.709-  ...........................  (14) 

=  1.857  j/^  .....  .  .  .  .  ...................  (15) 

=  .269      —  ..  ..(16) 

\  o 

t  =  50LX  —  .000127  c?L?  .............  .  ........  (17) 

T=t+  (R  +  o)  tan  J/  ........  ................  (18) 

E=(R  +  o)  exsec  J/  +  0  .....................  (20) 

C  =  100  L  —  .00034  a2L5  ......................  (21) 

u  =  x  —  y  cot  A 


,(22) 
=  ^r  —  v  cos  A  .  ...........................  j 

z;  =  _J^-  =  ^5  L  +  .000244  a2L5.  .  .  '.  (23) 


An  inspection  of  the  formulas  and  demonstrations  will 
show  the  following  properties  of  the  transition  spiral: 

28.  Degree-of-curve  The  degree-of-curve  at  any 
point  on  the  spiral  equals  the  degree  at  100  feet  from  the 
P.S.  multiplied  by  the  distance  along  the  spiral  from  the 
P.S.  to  the  point  (Eq.  1).  This  distance  must  be  ex- 
pressed in  units  of  100  feet  (stations).  Thus,  if  a  =  2,  at 
100  feet  from  the  P.S.  the  spiral  will  be  a  2°  curve;  at 
35  feet  (L  =  .25)  a  0°30'  curve;  at  450  feet,  (L  =  4.5)  a 
9°  curve.  ^~  is  the  number  of  feet  of  spiral  in  which 
D  changes  one  degree.  Thus,  for  a  =  2  the  spiral  increases 


SPIRAL   DEFLECTION    ANGLE  19 

its  degree  of  curve  one  degree  for  each  i-|-^=50  feet;  for 
a  =  §  one  degree  for  each  -!-££.  =  150  feet. 

At  the  terminal  point,  the  P.C.C.,  where  the  spiral  con- 
nects with  the  main  curve,  D  will  sometimes  be  repre- 
sented by  .Dj,  and  this  should  generally  equal  the  degree 
of  the  circular  curve  D0.  The  total  length  of  the  spiral 

will  be — -.      If  a  =  2,  a  6°  curve  would  require  a  spiral  3 
stations   (300  feet)   long. 

29.  Angle    A. — The  angle  A  between  the  initial  tan- 
gent  and   the   tangent  at   any   point   on   the   spiral    (the 
change    of   direction   corresponding   to   central    angle   of 
circular  curves)   (LTP,  Fig.  1,  page  5)  in  degrees  equals 
(Eq.  2)  : 

(a)  One  half  of  a  times  the  square  of  the  distance  in 
100- ft.  stations  from  the  P.S.  to  the  point;  thus  if  a  =  2, 
for  300  ft.  from  P.S.,  L  =  3,  and  A  =  i  x  2  x  32  =  9°.  Or 

(&)  One  half  of  the  product  of  this  distance  L  by  the 
degree-of-curve  of  the  spiral  at  the  given  point;  thus  at 
300  ft.  with  a  =  2,  D  =  6°,  and  A  =  i  x  3  x  6  =  9°.  Or 

(c)  One  half  of  the  square  of  degree-of-curve  at  the 
point  divided  by  a ;  thus  at  300  ft.  with  a  =  2,  A  ==  J  x  | f 
=  9°. 

For  the  same  angle,  then,,  the  spiral  is  twice  as  long 
is  a  circular  curve,  and  for  the  same  length  the  angle  is 
one  half  that  for  a  circular  curve  whose  D  is  the  same 
as  that  at  the  end  o^  the  spiral. 

30,  Spiral  deflection  angle  0.-     The  spiral  deflection 
angle  0  at  the  P.S.  from  the  initial  tangent  to  any  point 
on  the  spiral,  as  PAL  in  Fig.  1,  is  J  A,  or  J  a  L2.     Thus, 
for  a  point  300  ft.  from  the  P.S.  (L  =  3),  if  a  =  2,  0  =  4 
x2x  32  =  3°.   If  the  result  is  wanted  in  minutes,  since  Jx 


20  SUMMARY  OF  PRINCIPLES 

60  =  10,  use  10  instead  of  J.  For  105.4  ft.  with  a  =  2,  0 
=  10  x  2  x  (1.054) 2  =  22'.  0  is  also  one  third  of  the  deflec- 
tion angle  for  a  simple  curve  of  the  same  degree  as  the 
spiral  at  the  given  point.  Thus,  as  above,  the  deflection 
angle  for  300  ft.  of  6°  curve  is  9°  and  0  =  Jx9  =  3°. 

These  values  are  subject  to  slight  corrections  for  A 
larger  than  15°  or  20°  as  explained  in  the  derivation  of 
the  formula  on  page  9. 

31.  Tangent  at  point   on   spiral.       The     deflection 
angle  at  any  point  on  the  spiral  between  the  tangent  at 
this  point  and  the  chord  to  the  P.S.   (TLA  in  Fig.  1)  is 
A — 6.    This  enables  the  tangent  to  be  found.     For  A  less 
than  15°,  the  value  §  A  or  2  0  is  sufficiently  accurate. 
Thus,  for  the  preceding  example,  with  a  =  2,  for  the  point 
300  ft.  from  the  P.S.,  this  angle  is  20  =  6°. 

32.  Deflection   angle   at  point   on   spiral.      For  de- 
flection angles  from  a  point  on  the  spiral  to  other  points 
on  the  spiral,  the  principle  that  the  spiral  diverges  from 
the  osculating  circle   (circular  curve  of  same  degree)   at 
the   same   rate   that  the   spiral   deflects .  from   the   initial 
tangent  is  of  service.    The  angles  may  be  treated  in  three 
ways,  as  follows: 

33.  Angles   from  tangent.        By  equation    (10)   the 
deflection  angle  between  the  tangent  at  a  transit  point  on 
the  spiral  and  the  chord  to  any  other  point  on  the  spiral 
(as  CBH,  Fig.  4)  is  the  sum  or  difference  of  two  angles: 
(1)  the  deflection  angle  for  a  circular  curve  of  the  same 
degree  as  the  spiral  at  the  transit  point  for  a  length  equal 
to  the  distance  between  the  two  points,  and  (2)  the  spiral 
deflection  angle  0  for  a  lengh  of  spiral  equal  to  the  dis- 
tance between  the  two  points.     The  latter  angle. is  plus 
if  the  desired  point  is  farther  from  the  P.S.,  and  minus 
if  nearer,  than  the  point  from  which  the  deflections  are 
made. 


ANGLES  FROM   CHORD  21 

Thus,  if  a  =  2  and  the  transit  be  at  B  (Fig.  4),  250  ft. 
from  the  P.S.,  the  degree-of-curve  at  the  transit  point 
will  be  5°,  and  the  deflection  angle  CBH  to  set  a  point 
150  ft.  ahead  will  be  the  sum  of  3°  45',  (J  of  150  ft.  of 
5°  curve)  and  45',  (the  spiral  deflection  angle  for  150 


feet,  10x2x1.5)  or  4°30'.    For  D,  150  ft.  back,  it  would 
be  3°  45'  —  45'  =  3°0'. 

34.  Angles  from  chord. — Likewise  by  equation  (11) 
the  angle  CBE,  Fig.  4,  (deflection  angle  from  chord  to 
P.S.,)  may  be  calculated  by  adding  the  spiral  deflection 
angle  0  for  the  point  C  (GAC)  to  J  the  product  of  the. 
degree-of-curve  at  B  by  the  number  of  stations  from  the 
P.S.  to  C.  For  a  =  2  and  the  transit  at  B,  250  ft.  from 


FIG.  4 

the  P.S.,  the  degree-of-curve  at  the  transit  point  is  5°, 
and  the  angle  CBE  to  locate  the  point  C  150  ft.  ahead  and 
400  ft.  from  the  P.S.,  will  be  (J  x  2  x42  =  5°20')  +  (J  x  5 
x4=r3°  20')  .-=8°  40'.  For  the  point  D  100  ft.  from  the 
P.S,  the  angle  DBA  will  be  ( J  x  2  x  I2  ==  20')  +  ( J  x  5  x  1 
=  50/)=1°10'.  This  method  is  applicable  whether  the 
point  to  be  located  is  nearer  to,  or  farther  from,  the  P.S. 
than  the  transit  point.  It  permits  the  calculation  of 
the  spiral  deflection  angles  at  P.S.  for  the  whole  spiral 
and  the  determination  of  the  angles  between  the  chords 
in  question  by  adding '  to  these  spiral  deflection  angles 


22  SUMMARY    OF    PRINCIPLES 

the  angles  J  D'  L,  where  D'  is  the  degree-of  curve  at  the 
transit  point  and  L  is  the  distance  from  P.S.  to  the  point 
to  be  located. 

35.  Angles  with   initial    tangent. — Equation     (12) 
gives  the  deflection  angles  from  a  line  parallel  with  the 
initial  tangent.     The  results  are  the  same  as  if  the  spiral 
deflection   angle  ff  for  the  transit  point  were   added  to 
those  from  the  chord  found  in  the  preceding  paragraph. 

36.  Divergence  from  osculating  circle. — The    spiral 
diverges  from  its  osculating  circle  (circular  curve  of  the 
same   degree)    at   any  point   at  the   same   rate   that   the 
spiral  deflects  from  the  initial  tangent,  and  the  distance 
between  the  circle  and  spiral  is  the  same  as  the  y  for 
an  equal  length  of  spiral. 

This  enables  the  spiral  to  be  located  by  offsets  measured 
from  the  circular  curve.  By  this  method  half  of  the 
spiral  may  be  located  from  the  initial  tangent  and  half 
from  the  produced  circular  curve,  the  offsets  for  the  two 
being  the  same  for  the  same  distances  from  the  P.S.  and 
the  P.C.C.  respectively.  See  Fig.  5. 

37.  Abscissa  fie. — The  distance  in  feet  along  the  ini- 
tial tangent  to  the  perpendicular  to  a  point  on  the  spiral 
is  equal  to  the  length  along  the   spiral  in  feet  less  the 
quantity  .000762  a2  L5,  where  L  is  the  length  along  the 
spiral  expressed  in  units  of  100  ft.    A  convenient  way  to 
find  x  is  to  have  this  quantity  tabulated  for  given  spirals 
as  a  correction,  or  it  may  easily  be  found  from  tabulated 
values  of  such  a  correction  for  a  =  1  by  multiplying  by  a2. 
For  extreme  lengths  another  term  may  be  needed.     As 
a»n  illustration,  with  a  =  1,  for  200  ft.  the  correction  to  be 
subtracted  from  200  ft.  to  find  x  is  .000762  x  32  =  .02  ft., 
a  small  quantity 

38.  Ordinate   y. — The  ordinate  y  (perpendicular  dis- 
tance from  the  initial  tangent  to  a  point  on  the  curve) 


OFFSET  23 

in  feet  equals  .291  times  the  product  of  a  by  the  cube  of 
the  distance  along  the  spiral  from  the  P.S.  to  the  point 
expressed  in  units  of  100  ft.  (stations).  It  therefore 
varies  as  the  cube  of  the  distance  from  P.  S.  Knowing  y 
for  one  point,  the  y  for  a  second  point  may  be  computed 
from  it  by  this  relation.  D  L2  may  be  substituted  for 
a  L3.  For  extreme  lengths,  a  third  term  may  have  to  be 
considered.  As  an  illustration,  with  a  =  1  for  200  ft 
(L  =  2)  ,  y  =  .291  x  8  =  2.33  ft.  For  100  ft,  y  is  one  eighth 
as  great;  for  400  ft.  y  may  be  used  as  eight  times  as 
great,  though  the  use  of  the  next  term  of  the  series  would 
change  this  somewhat. 

39.  Offset  o.  —  The  offset  o  between  the  initial  tan- 
gent and  the  parallel  tangent  from  the  main  curve  pro- 
duced backward,  (BD,  Fig.  1),  in  feet  equals  .0727  times 
the  product  of  a  by  the  cube  of  the  length  of  the  whole 
spiral  in  stations,  or  .0727  times  the  square  of  the  length 
of  spiral  and  the  degree  of  main  curve.    This  ordinate  is 
approximately  one  fourth  of  the  ordinate  y  of  the  end  of 
spiral.     The  spiral  bisects  the  offset  at  a  point  half-way 
between  the  P.S.  and  the  P.C.C.     (Eq.  11.)     BE  =  ED. 
AE  =  EL.     (Fig.  1.)     The  slight  error  in  this  is  discussed 
in  the  derivation  of  the  formulas  (page  14).     The  value 
of  o  may  best  be  discussed  by  means  of  one  of  the  tables. 

40.  Calculation   from   known   values.  —  When     the 
length  of  the  spiral  is  not  so  great  that  a  second  or  cor- 
rection term  is  needed  for  the  values  of  0,  y,  A,  etc.,  it 
is   seen    from   equations    (9),    (6),    (2),   etc.,   that  these 
functions   vary  as  the  square  and  cube  of  the  distance 
L  and  may  be  calculated  from  any  known  value.     Thus 


r2 

if  0  for  400  ft.  is  2°40',  for  300  ft,    #=  —  2#i  ={  -  \     X 

£1          [400J 

(2°  400=1°  30'.     If  y   for  400  ft.   is   18.59,   for  300   ft. 


24  THE  TABLES 

T  3         fsool 3 

y  =-7-5  j/i,  =  \  -  -  \  x  18.59  =  7.85.     The  deflection  angle 
£i  [400J 

varies  as  the  square  of  the  distance  and  the  ordinate  as 
the  cube  of  the  distance  from  the  P.S. 

41.  t  and  C. — The  distance  t  from  the  P.S.  to  this 
offset  (AB;  Fig.  1)  is  found  by  subtracting  the  correction 
.000127  a2  L±5  from  the  half  length  of  the  curve  in  feet. 
(Eq.  17.)     Generally  this  correction  term  is  quite  small. 
As  stated  on  page  13  this  term  may  be  tabulated,  and  it 
may  also  be  obtained  for  a  given  length  of  spiral  by  mul- 
tiplying tabulated  values  for  a  =  1  by  the  square  of  the  a 
of  the  given  spiral.    For  this  use  see  pages  25  and  26. 

The  long  chord  C  is  found  by  subtracting  the  correc- 
tion, .000338  a2  L5,  from  the  length  of  the  curve  in  feet. 
(Eq.  21.)  This  correction  may  be  found  by  multiplying 
the  x  correction  for  the  same  length  of  spiral  by  four 
ninths 

42,  u  and  v. — The  spiral  tangent-distances  u  and  v 
(AT  and  TL,  Fig.  1)   are  found  by  equations   (22)   and 
(23).    v  can  be  found  most  easily  by  taking  one  third  of 
the  tabulated  values  of  the  x  correction  and  adding  this 
to  one  third  of  the  length  of  the  spiral  in  feet. 


THE  TABLES 

43.  The  computations  may  be  shortened  by  the  use 
the  tables. 

Tables  I-XI  gives  the  values  of  the  principal  parts  of 
the  transition  spiral  for  the  following  values  of  a:  J,  §, 
f,  1,  1J,  If,  2,  2J,  3J,  5,  and  10.  The  column  headed 
"Length"  is  the  distance  in  feet  along  the  spiral  from 
the  P.S.  to  any  point  on  the  spiral,  and  is  equal  to  100 
times  the  L  of  the  formulas.  The  column  headed  "x  COR." 


INTERPOLATION  25 

gives  the  correction  to  be  subtracted  from  this  distance 
in  feet  along  the  spiral  to  obtain  x,  and  that  headed 
"t  COR."  gives  the  correction  to  be  subtracted  from  the 
half  length  of  the  spiral  in  feet  to  obtain  t.  Both  t  COR. 
and  o  are  to  be  taken  from  the  line  for  the  full  length  of 
the  spiral.  For  example,  by  Table  IV,  with  a  =  l,  to  con- 
nect with  a  5°  curve,  the  length  of  spiral  is  500  ft.  and 
L  =  5;  the  change  of  direction  A;  is  12°  30';  the  offset  o 
to  P.C.  of  circular  curve  is  9.07  ft;  t  is  250  —  A  =  249.6 
ft.;  x  is  500  —  2.37  =  497.63  ft.;  and  the  values  of  A  A,  6, 
yy  and  x  COR.  for  points  200,  210,  220  ft.,  etc.,  distant  from 
the  P.S.  are  found  in  the  line  with  200,  210,  etc. 

To  find  the  long  chord  to  P.S.  C,  subtract  .445  of  x  COR. 
from  the  length  of  the  curve  in  feet.  To  find  the  spiral 
tangent-distance,  v,  add  one  third  of  x  COR.  to  one  third 
of  the  length  of  the  spiral  in  feet. 

Tables  I-IV  have  the  values  of  A  and  0  calculated  to 
the  nearest  tenth  of  a  minute,  and  Tables  V-VII  to  the 
nearest  half  minute.  While  this  precision  is  not  usually 
necessary,  it  may  be  of  service  where  the  sum  of  two  or 
more  angles  is  used. 

44.  Interpolation.  — To  find  values  intermediate  be- 
tween the  distances  given  in  the  tables,  interpolate  by 
multiplying  one  tenth  of  the  difference  between  consec- 
utive values  by  the  number  of  additional  units.  Thus, 
Table  IV  gives  A  for  400  ft.  as  8°  00';  for  410  ft.,  8°  24'.3. 
One  tenth  of  the  difference  between  these  is  2'.4.  For 
406.8  ft,  add  6.8  x  2.4  =  16'.3  to  8°00',  giving  8°16'.  For 
y,  add  6.8  x. 143  =  .97  to  18.59,  giving  19.56  ft  For  o, 
add  6.8  times  one  tenth  of  .36  to  4.65  giving  4.89.  D  is 
4.068  or  4°  4'.08. 

Interpolation  may  also  be  made  for  other  columns. 
Thus  if  o  is  given  as  7.0  ft.  and  a  =  lj,  by  Table  V  the 
length  of  spiral  will  be  between  420  and  430  ft.  Inter- 
polating, as  o  increases  0.5  ft.  in  10  ft.  of  length,  the  .28 


26  THE  TABLES 

will  be  gained  in  5.6  ft.  and  the  length  is  425.6  ft.  Inter- 
polation for  A,  D,  etc.,  may  then  be  made  as  before. 
Again,  for  D  =  4°  16',  still  using  a  =  1 J,  the  length  is  be- 
tween 340  and  350  and  is  ^x  10  =  1.33  ft.  more  than  340, 
making  341.33. 

In  general  this  interpolation  gives  accurate  results  and 
no  correction  need  be  made.  For  A  the  error  in  inter- 
polation with  values  of  a  greater  than  5  may  need  to  be 
taken  into  account.  To  find  exact  values  of  A,  deduct 
from  the  interpolated  values  a  times  the  following  quan- 
tities: For  a  length  in  feet  ending  with  1,  .027';  2,  .048'; 
3,  .063';  4,  .072';  5,  .075';  6,  .072';  7,  .063';  8,  .048';  9, 
.027'.  It  can  easily  be  determined  whether  this  correc- 
tion need  be  considered.  The  difference  arises  from  the 
fact  that  the  square  of  numbers  does  not  increase  uni- 
formly. For  the  other  columns  the  errors  of  interpolation 
are  very  slight  and  may  be  neglected. 

45.  General  use  of  Table  IV. — Table  IV  has  been 
carried  to  several  decimal  places  to  permit  its  use  for 
values  of  a  other  than  1.  To  calculate  values  for  another 
a,  multiply  the  tabular  value  of  D,  A,  0,  o,  or  y  in  Table 
IV  for  the  distance  from  the  P.S.  to  the  point  on  the 
spiral  by  the  a  of  the  spiral  used,  and  the  x  COR.  and  t 
COR.  by  the  square  of  the  a  of  the  spiral.  Thus  if  a  =  2.2 
and  L  =  3.1,  multiply  the  D,  A,  0,  o,  and  y  opposite  310  by 
2.2,  and  the  x  COR.  and  t  COR.  by  the  square  of  2.2.  The 
values  of  y,  o,  and  x  COR.  obtained  in  this  way  are  subject 
to  slight  errors  for  large  values  of  a  if  A  is  more  than 
18°,  but  fortunately  y  for  a  distance  greater  than  half  of 
the  length  of  the  spiral  is  seldom  needed,  and  as  the 
error  of  this  and  the  errors  in  o  and  x  COR.  are  ordinarily 
small  the  correction  may  generally  be  neglected.  The 
amount  of  this  error  may  be  found  by  the  method  given 
in  a  succeeding  paragraph.  The  error  in  0  is  discussed 
on  page  9. 


CORRECTIONS  27 

To  use  Table  IV  for  another  a,  it  may  be  desirable 
first  to  determine  the  length  of  the  spiral  by  dividing  the 
DI  of  the  required  spiral  by  a  or  to  determine  it  from  o. 
Thus,  for  a  =  1.5,  to  connect  with  a  6°  curve,  divide  6 
by  1.5,  which  gives  L±  =  4;  that  is,  the  whole  spiral  will 
be  400  ft.  long,  and  the  properties  for  the  spiral  may  be 
computed  by  multiplying  those  in  the  line  with  the  re- 
quired distance  by  1.5.  In  other  words  it  must  be  borne 
in  mind  that  the  distances  in  the  column  of  lengths 
remain  unchanged  with  new  values  of  a,  and  the  quan- 
tities in  all  the  other  columns  will  be  changed  for  a  other 
than  1 

46.  Corrections  for  calculations. — For  the  calcula- 
tion of  tables  and  other  work  requiring  the  recognition 
of  a  further  term  in  the  equations,  the  value  of  the  sec- 
ond term  of  the  o  series  (.0000002  a3  L\  eq.  (13)  )  and 
of  the  second  term  of  the  y  series  (.00000158  a3  U,  eq. 
(6)  )  may  be  obtained  by  multiplying  the  quantities  in 
the  following  table  by  a3;  and  the  third  term  of  the  x 
series  (.00000000268  a4  L°,  eq.  (7)  )  by  a4.  These  terms 
for  o  and  y  are  negative,  and  the  term  fo«  x  is  to  be 
subtracted  from  the  x  COR. 

L  o  y                        x 

2.50  .0010  

3.00  .0004  .0035  

3.50  .0013  .010  

4.00  .0032  .026  .0007 

4.50  .0074  .059  .002 

5.00  .015  .124  .005 

5.50  .030  .241  .012 

6.00  .055  .442  .027 

6.50  .097  .775  .055 

7.00  .163  1.301                1.08 
For  making  corrections  on  results  obtained  from  Table 
IV   for   a   other   than   1,    subtract   from   the   product   of 


28  CHOICE  OF  SPIRAL 

the  multiplication  used  to  obtain  the  desired  distance 
a  (a2  —  1)  times  the  value  from  the  above  table  in  obtain- 
ing o  and  y,  and  a  (a3  —  1)  times  the  value  from  the  table 
in  obtaining  the  x  COR. 

47.  Table  of  ordinates. — By  Table  XII  the  ordinate 
from  the  tangent  or  from  the  circular  curve  at  a  decimal 
part  of  the  half  length  of  the  spiral  may  be  obtained  by 
the  multiplication  of  o  of  the  spiral  by  the  factor  given 
in  the  table.     See  method  by  co-ordinates  and  Fig.  5.     It 
should  not  be  forgotten  that  Tables  I-X  give  ordinates, 
and   that   values    for   intermediate   points   may   easily   be 
interpolated. 

48,  Table  of   offsets. — Table   XX   gives   values   of  o 
and  L  for  various  values  of  a.    Within  reasonable  limits 
it  will  bear  interpolation,  both  for  intermediate  values  of 
a  and  D  and  to  determine  a  for  intermediate  values  of  o. 
It  is  of  service  in  location  problems. 

Tables  XIII  and  XIV  are  described  under  Uniform 
Chord  Length  Method.  The  tables  for  street  railway 
curves  are  described  under  Street  Railway  Spirals. 


CHOICE  OF  a  AND  LENGTH  OF  SPIRAL. 

49.  The  selection  of  a  and  with  it  the  length  of  spiral 
require  consideration.  The  value  of  a  to  be  used  is 
dependent  upon  the  speed  of  trains,  the  maximum  degree- 
of-curve,  the  length  of  tangents,  the  permissible  offset 
of  the  line  for  the  topographical  conditions  in  question, 
the  distance  in  which  the  superelevation  of  the  outer 
rail  may  be  attained,  etc.,  and  hence  is  subject  to  a  wide 
range  of  conditions.  It  may,  however,  aid  the  engineer's 
judgment  to  discuss  these  conditions  briefly. 

I 


ATTAINMENT  OF  SUPERELEVATION  29 

50.  Effect   of   speed.     -For  the  same  rolling  stock  and 
for  the  same  comfort  in  riding,  it  would  seem  that  a  given 
amount  of  superelevation  must  be  attained  in  the  same 
length  of  time ;  and  hence  it  is  probable  that  a  should  vary 
nearly  inversely  as  the  cube  of  the  speed  of  the  train. 
This  conclusion  also  emphasizes  the  desirability  of  spiral- 
ing  curves  used  under  high  speeds. 

Assuming  that  a  =  l  is  a  proper  value  for  speeds  of 
50  miles  per  hour,  this  principle  would  suggest  the  follow- 
ing maximum  values  of  a ;  60  miles  per  hour,  J ;  50  miles 
per  hour,  1;  40  miles  per  hour,  2;  30  miles  per  hour,  3J; 
25  miles  per  hour,  5;  20  miles  per  hour,  10.  While  for 
the  very  high  speeds  this  may  seem  to  require  unneces- 
sarily long  spirals  and  for  low  speeds  short  spirals,  yet 
a  =  1  has  given  satisfactory  results  at  speeds  of  50  to  60 
miles  an  hour,  and  a  =  2  at  40  to  50  miles  an  hour,  and 
for  60  miles  an  hour,  a  =  J  is  not  too  small.  Of  course, 
in  any  case,  longer  spirals  and  smaller  values  of  a  will 
give  smoother  riding  curves. 

51.  The  speed  of  trains  may  be  limited  by  the  maxi- 
mum   superelevation    allowable    on    the    sharper    curves.' 
Under  usual  practice  the  requirement  of  maximum  super- 
elevation would  limit  the  maximum  degree-of-curve  for 
speeds  of  60  miles  an  hour  to  3°,  for  50  miles  to  4°,  for 
40  miles  to  6°,  for  30  miles  to  12°,  etc.    Where  the  track 
is  not  used  for  slow  trains  and  a  superelevation  of  more 
than  7  or  8  inches  is  allowable,  somewhat  higher  speeds 
on  such  curves  may  be  used.     The  maximum  speed  of 
train,  however,  will  be  the  governing  consideration  in  the 
choice  of  a  rather  than  the  maximum  degree-of-curve. 

52.  Attainment    of    superelevation. — The    rate    of 
attaining  the   superelevation   is   sometimes   given   as   the 
governing  consideration,  but  in  reality  this  rate  is  gov- 


30  CHOICE  OF  SPIRAL 

erned  by  the  speed.  The  distance  in  which  the  outer 
rail  should  attain  an  elevation  of  1  inch  will  not  be  the 
same  for  a  speed  of  60  miles  an  hour  as  for  one  of  -iO 
miles.  The  schedule  of  maximum  values  of  a  for  various 
speeds  as  given  above  involves,  approximately,  attaining 
1  inch  of  elevation  in  the  following  distances :  60  miles, 
80  feet;  50  miles,  53  feet;  40  miles,  44  feet;  25  miles,  40 
feet.  The  best  rate  also  depends  upon  stiffness  of  car 
springs,  weight  and  style  of  cars,  and  other  'Conditions. 
Generally  speed  and  amount  of  superelevation  should  gov- 
ern the  length  of  spiral,  and  rate  of  attainment  is  sub- 
ordinate. 

53.  It    may    be    convenient    for    maintenance-of-way 
work  to  arrange  the  spiral  so  that  the  superelevation  is 
attained  at  a  definite  rate  per  100  ft.  of  length  of  spiral. 
Let  k  be  this  rate,  expressed  in  inches  of  superelevation 
attained  in  100  feet.    Let  h  be  the  superelevation  in  inches 
per  degree  of  curve;  for  a  3°  curve,  3h,  etc.     Then 

,          ,        D  .  k 

k  =  ah  ~  —~h.  a  =  —r. 

L  h 

54.  The  following  table  shows  the  values  of  0,  which 
gives  rates  of  1,  1J  and  2  inches  of  superelevation  attained 
in  100  ft.  for  the  amount  of  superelevation  per  degree  of 
curve  given  at  the  head  of  the  columns. 


VALUES  OF  a 

Elevation  per  degree  J         J       1       li     U     2       2J 

a  for  k  equal  to  1     in.  per  100  ft.  2  1%  1           i  %        %        f 
a  for  k  equal  to  IK  in.  per  100  ft.  3         2  \%    .   f  1  K        f 
a  for  k  equal  to  2     in  per  100  ft.  4  2%  2         if  1%      1           I 
Velocity  in  miles  an  hour  corres- 
ponding to  superelevation  26.9  33.0  38.1     42.6  46.6     53.8     60.2 

The  length  of  spiral  for  a  =  J  is  200  ft.  for  each  degree 
of  curve ;  f  or  a  =  |,  125  ft. ;  for  a  =  §,  150  ft.,  etc.    When 


MINIMUM  SPIRALS  31 

tables  are  not  given  for  the  a  used,  the  values  may  be 
calculated  from  the  tables  by  multiplication  or  other 
process.  Thus  for  a  =  lj,  double  the  values  for  a  =  §; 
for  a  =  i,  multiply  those  from  a  =  J  by  1J  or  those  from 
a  =  1  by  f . 

55.  The  amount  of  superelevation  per  degree  of  curve 
here  used  is  calculated  from  .00069   F2,  where  V  is  the 
velocity   in  miles  per  hour.     This   gives   the   number  of 
inches  per  degree  to  counteract  the  centrifugal  force,  and 
is  based  on  distance  from  center  to  center  of  rail.     This 
amount  is  used  here  because  it  is  a  value  quite  commonly 
quoted;  for  other  superelevations  comparisons  may  read- 
ily be  made  with  the  figure  here  given.     There  is  a  wide 
divergence  of  opinion  on  the  proper  amount.     A  super- 
elevation  somewhat  less  than  that   required  to  counter- 
balance the  centrifugal  force  produces  a  moderate  flange 
pressure  on  the  outer  rail  and  is  held  by  many  to  give 
a  smoother  riding  track.     Care  must  be  taken  that  track 
so   elevated   is   never   used   at   speeds   so   far   above   the 
assumed  speed  as  to  be  unsafe.    For  convenience  in  main- 
tenance-of-way  work  it  may  be  desirable  to  establish  the 
superelevation  at  a  convenient  amount  near  the  value  cal- 
culated for  the   assumed   velocity,   an  allowable  practice 
since  the  assumed  velocity  may  not  be   realized.     Thus 
2  inches  per  degree  may  be  used  in  place  of  If,  etc. 

56.  Minimum  spirals.    For  a  given  value  of  a  there 
may  be  a  question  as  to  how  flat  a  curve  may  profitably 
be    spiraled.      The   spiral    should    certainly   vary   enough 
from   the   position   of   a   simple   circular   curve   that   the 
distinction  may  not  be  obliterated  by  the  inaccuracies  of 
track  work;  otherwise  it  will  be  as  advantageous  to  begin 
the  superelevation  an  equal  distance  back  on  the  tangent. 
The  limits  given  in  a  former  edition  have  been  criticised 
by  engineers  of  maintenance  of  way,  and  experience  on 


32  CHOICE  OF   SPIRAL 

prominent  roads  indicates  that  the  minimum  limit  of  o 
there  set,  0.6  to  1  ft.,  was  too  high.  It  seems  that  the 
gradual  change  of  direction  between  the  trucks  and  the 
car  body,  and  the  fitting  of  elevation  to  the  curvature 
by  the  spiral,  are  advantageous  to  -smooth  riding  even 
when  the  change  in  alignment  is  slight.  Experience 
seems  to  indicate  that  for  a  =  J  curves  above  30'  may  be 
spiraled;  for  a  =  l,  1°  and  above;  for  a  =  2,  2°;  for  a  = 
3J,  3°;  for  a  =  5,  4°;  for  a  ^  10,  6°.  For  curves  lighter 
than  these  any  advantage  seemingly  found  by  spiraling 
would  probably  be  obtained  by  beginning  the  supereleva- 
tion back  on  the  tangent. 

In  any  case  decreasing  the  value  of  a  and  thus  increas- 
ing the  length  of  the  spiral  will  increase  the  efficiency 
of  the  spiral  and  better  the  riding  qualities  of  the  curve. 
This  view  needs  emphasizing,  and  too  much  should  not 
be  expected  of  short  spirals. 

57.  Selection  of  a  and  length  of  spiral — The  selec- 
tion of  a,  then,  must  be  a  matter  to  be  left  to  the  judg- 
ment of  the  engineer.  As  a  guide  the  following  table 
containing  values  of  a  which  have  given  satisfactory 
results  at  the  speeds  noted  is  given.  Lower  values  of  a 
are  of  course  advantageous;  as,  for  example,  at  a  speed 
of  40  miles  an  hour  a  =  lj,  or  even  1  will  make  a  more 
efficient  easement  than  the  one  given.  Higher  values  of 
a— shorter  spirals — may  be  necessary  in  many  cases,  but 
it  must  be  understood  that  they  will  not  be  so  satisfactory. 
The  column  headed  "Minimum  curve  spiraled"  is  the 
lightest  curve  which  it  is  considered  desirable  to  spiral 
with  the  value  of  a  given  opposite.  "Maximum  curve" 
is  fixed  at  the  given  speed  by  the  limit  of  superelevation; 
at  lower  speeds  this  a  may  profitably  be  used  for  sharper 
curves.  The  speeds  are  given  in  miles  per  hour  and  the 
elevations  in  inches 


P.S..  P.C.C.,  AND  P.C.  33 

MINIMUM  SPIRAL  FOR  MAXIMUM  SPEED 

Maximum  s.  Maximum    Min.  Curve    Length  per    Elev.  per 

Speed  ^  Curve  Spiraled  Degree        Degree 

60  J  3°  30'  200  2-1 

50  1  4°  1°  100  15 

40  2  7°  2°  50  1J 

30  3J  11°  3°  30  f 

25  5  14°  .        4°  20  J 

20  10  25°  5°  10  ^ 

For  shorter  spirals,  the  following  values  of  a  are  con- 
sistent with  each  other :  60  miles  per  hour,  1 ;  50  miles 
per  hour,  1|;  40  miles  per  hour,  3J;  30  miles  per  hour, 
6| ;  25  miles  per  hour,  10. 


LOCATION  OF  P.S.,  P.C.C.,  AND  P.C. 

58.  Location  from  intersection  of  tangents. — When 
the  tangents  have  been  run  to  an  intersection,  the  P.S. 
(A,  Fig.  1,  page  5)  may  be  located  by  measuring  back  on 
the  tangent  from  the  point  of  intersection  a  distance 
equal  to  the  tangential  distance  T  (equation  18).  This 
distance  may  also  be  computed  by  adding  t  +  o  tan  J/  to 
the  tangential  distance  of  the  circular  curve  as  ordinarily 
calculated.  (See  section  21.).  The  P.C.  (D,  Fig.  1) 
may  be  located  by  calculating  o  and  offsetting  this  amount 
at  a  point  on  the  tangent  distant  t  from  the  P.S.  (See 
section  19.).  t  may  be  found  by  subtracting  the  t  COR. 
of  the  tables  from  the  half  length  of  curve  in  feet.  Thus, 
for  400  ft.  of  spiral  with  a  =  2,  by  Table  VII  t  cor.  is 
.5  ft.  and  £  =  200  —  0.5  =  199.5  ft.  The  P.C.C.  (L,  Fig.  1) 
may  then  be  located  by  running  the  spiral  from  the  P.S., 
or  by  locating  the  circular  curve  from  the  P.C.  for  a 
distance  jL±. 


34  PSV  P.C.C.,  AND  P.C. 

59.  Location  from  P.C.  of  a  curve  without  spiral.    In 

case  a  simple  curve  has  been  run  without  provision  for 
a  spiral  and  without  offsets,  that  is  in  the  usual  way, 
it  will  be  necessary  to  change  the  position  of  the'  circular 
curve.  The  distance  of  the  P.S.  back  of  the  P.C.  of  the 
old  simple  curve  will  be  t  •+  o  tan  J  I,  I  being  the  total 
intersection  angle.  The  new  curve  will  come  inside  the 
old  but  will  not  be  exactly  parallel  to  it. 

60.  Location   from  P.  C.   of   offsetted  curve.      If  a 

simple  curve  has  been  run  for  use  with  spiral,  as  DLH  in 
Fig.  1,  page  5,  o  may  be  computed,  the  offset  measured 
to  B  and  the  distance  t  (AB)  measured  to  locate  the 
P.S.  (A).  The  length  J  L±  measured  from  the  P.C.  on 
the  circular  curve  will  locate  the  P.C.C.  (L).  Similarly 
if  the  tangent  is  fixed,  the  curve  may  be  located  by  first 
making  the  offset  from  the  tangent  to  the  P.C. 

61.  If  botk  P.C.  and  tangent  are  fixed  with  an  offset 

np 
o  =  BD  between  them,  a  may  be  found  from  a  =  .269  .  — 

or  a  and  L  may  be  found  from  Table  XX.  After  finding 
t,  the  P.S.  may  be  located  in  the  usual  manner.  For  a 
5°  curve  with  0=10  ft.,  by  equations  (14)  and  (16) 
L  =  525.3  ft.  and  a  =  .952.  Since  with  a  =  l  and  this 
length  of  spiral  t  cor.  =  .5,  the  correction  to  be  used  here 
is  0.5  x  a2,  and  t  =  262.65  —  .45  =  262.2  ft.  This  method 
is  a  great  convenience  where  it  is  desired  on  account  of 
the  ground  to  throw  the  curve  in  or  out  without  changing 
tangent,  or  where  a  similar  change  in  the  tangent  is  de- 
sired without  a  change  in  the  curve,  the  connection  to  be 
made  by  means  of  a  suitable  spiral. 


LOCATION   BY  CO-ORDINATES  35 


LAYING  OUT  THE  SPIRAL  BY  CO-ORDINATES. 

62.  With   the   initial   tangent   as   axis   of   X   and  the 
P.S.   as  the  origin  of  co-ordinates,   it  is  not  difficult  to 
locate   points    on    the    spiral    by    means    of    co-ordinates. 
These  may  be   calculated   from  equations    (6)    and    (7), 
or   they  may  be  taken    from   the   tables.     Beyond   B   of 
Fig   1,   page   5,   the   ordinates   become   large   and   the  x 
correction   may   be   considerable.      For  long   spirals,    the 
second  term  of  the  3;  series,  may  need  to  be  considered. 
The  property  that  the  spiral  diverges  from  the  circular 
curve  at  the  same  rate  as  from  the  initial  tangent  is  of 
service.     Between   E   and   the   P.C.C.    (L)    measure  the 
ordinate  or  offset  from  the  circular  curve,  using  for  this 
offset  at  a  point  a  given  distance  from  the   P.C.C.  the 
ordinate  y  of  the   spiral   from  the   tangent  at  the   same 
distance  from  the  P.S.     Thus  for  a=l,  by  Table  I,  for  a 
point  200  ft.  from  the  P.S.,  y  =  2.33  ft.      To  locate  a  point 
on  the  spiral  200  ft.  from  the  P.C.C,  offset  from  the  cir- 
cular curve  this  same  distance,  2.33  ft. 

63.  Knowing  the  y  for  any  point  on  the  curve,  the  y 
for  any  other  point  within  ordinary  limits  may  be  found 
by   multiplying  the   former  number  by  the   cube   of   the 
ratio   of  the   distances   from  the   P.S.   to  the   respective 
points;  thus,  if  for  a  point  250  ft.  from  the  P.S.,  y  =  4.54 
ft.,  y  at  300  ft.  equals  (  ||  )3x  4.54  =  7.85  ft.     Similarly, 
points  may  be  located  by  offsets  from  the  circular  curve, 
the  distance  being  measured  from  the  P.C.C. 

64.  If  the  length  of  the  half  spiral  be  divided  into  an 
integral  number  of  parts  (See  Fig.  5),  any  ordinate  from 
the   tangent   or   from   the   circular   curve   may   easily  be 
calculated  from  0(0  =  .0727  DL\  pp.  22  and  25),  by  mul- 


36 


LOCATION   BY   CO-ORDIXATES 


tiplying  o  by  one  half  the  cube  of  the  ratio  of  tne  number 
of  parts  this  point  is  from  the  P.S.  to  the  whole  number 
of  parts.  The  following  table  gives  the  factor  by  which 
the  o  of  the  spiral  may  be  multiplied  to  determine  the  y 
of  the  point  when  the  length  of  the  half  spiral  is  divided 
into  10  parts. 


RS. 


1 
0 


RC. 


FIG.  5 


RC.C 


TABLE  OF   FACTORS   FOR   ORDIXATES. 

To  find  y,  multiply  o  by  the  factor. 


Ratio  to  half 
length. 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0  7 

0.8 

0.9 

1.0 

Factor. 

.0005 

.004 

.014 

.032 

.063 

.108 

.172 

256 

.365 

.500 

As  an  example,  \vith  a  =  l,  o  for  a  5°  curve  (spiral  500 
feet  long)  is  9.07  ft.  The  half  length  of  the  spiral  is 
250  ft.  and  one  tenth  of  this  distance  is  25  ft.  y  at  100  ft. 
(0.4  of  the  half  length)  is  .032  x  9.07  =  .29  ft.  Similarly, 
.29  ft.  will  be  the  offset  from  the  circular  curve  at  a  point 
100  ft.  from  the  P.C.C. 

To  divide  the  half-length  of  spiral  into  5  parts  and 
the  whole  spiral  into  10  parts,  use  the  even  numbered 
tenths  of  the  above  table. 


LOCATION  FROM   SPIRAL  TANGENTS  37 

For  intermediate  values,  interpolation  in  the  above 
table  will  give  reasonably  accurate  results.  This  enables 
interpolation  for  quarter  points,  and  other  fractional 
parts;  thus,  for  .67  of  half  length  the  factor  is  .153. 

The  results  by  the  above  table  are  subject  to  error  in 
the  hundredths  place,  but  for  usual  cases  are  within  .02  ft. 

65.  Still  another  method  is  to  measure  ordinates  from 
the  initial  tangent  for  about  two  thirds  of  the  length  of 
the  spiral  and  for  the  remaining  distance  to  measure  ordi- 
nates or  offsets  from  the  terminal  spiral  tangent  (TL, 
Fig.  1,  page  5).  The  offsets  from  the  terminal  spiral 
tangent  will  be  the  difference  between  the  offset  for  the 
produced  circular  curve  DL  and  the  y  for  a  spiral,  both 
for  a  length  equal  to  the  distance  from  P.C.C.  to  the  point 
to  be  located.  As  this  distance  will  be  less  than  one  third 
the  spiral  length,  the  approximate  formula  for  tangent 
offset,  .873  Dj  L2,  may  ordinarily  be  used.  D^  is  the 
degree  of  the  spiral  at  the  P.C.C.  and  L  here  will  be  used 
as  the  distance  from  the  P.C.C.  to  the  desired  point.  The 
offset  will  then  be  .873  D^  L2— .291  a  L3.  While  the  off- 
sets are  longer  than  those  from  the  circular  curve,  the 
measurements  are  made  from  the  tangent  and  the  circular 
curve  need  not  be  run.  As  an  example  take  a  =  1,  and 
£^  =  4°.  Length  of  spiral  is  then  400  ft.  For  a  point 
50  ft.  back  of  P.C.C.  (L  =  .5)  the  offset  is  .87  — .04  — .83 
ft.  For  100  ft.  from  P.C.C.  the  offset  is  3.49  —  .29  =  3.20 
ft.  This  is  a  very  convenient  method. 

66.  Many  engineers  prefer  the  co-ordinate  method. 
The  circular  curve  is  run  from  the  P.C.  established  by 
making  the  offset  from  the  initial  tangent,  and  the  spiral 
is  then  located  by  setting  off  ordinates  from  the  simple 
curve  between  P.C.  and  P.C.C.  and  by  ordinates  from  the 
initial  tangent  back  to  the  P.S.,  or  for  the  latter  portion 


38  LOCATION  BY  TRANSIT  ANGLES 

by  laying  off  o — y  from  a  tangent  at  the  P.C.  parallel  to 
the  initial  tangent,  the  ordinates  being  calculated  by  one 
of  the  preceding  methods;  or  offsets  from  the  terminal 
spiral  tangent  may  be  made  for  the  last  third  of  the  spiral 
length.  This  method  is  particularlly  applicable  to  location 
work  and  to  short  spirals,  though  under  many  conditions 
it  may  readily  be  applied  to  setting  track  centers. 


LAYING   OUT   THE   SPIRAL   BY   TRANSIT   AND 
DEFLECTION  ANGLES. 

67.  The  spiral  may  be  run  in  with  the  transit  by  turn- 
ing off  deflection  angles  and  making  measurements  along 
chords  in  much  the  same  manner  as  circular  curves.    The 
deflection  angles  are  easily  calculated,  and  the  field  work 
is  not  more  difficult  than  for  circular  curves.     The  ordi- 
nary transit-man  will  find  no  difficulty  in  understanding 
the  work.     Since  it  is  not  necessary  to  keep  succeeding 
chords  the  same  length  as  the  first,  the  stationing  may 
be  kept  up,  and  the  even  stations,  -|-50's,  and  other  points 
put  in  as  usual.     Herein  is  an  advantage  over  methods 
requiring  a  regular  length  of  chord  to  be  used. 

68.  Transit  at  P.  S.     With   the   transit    at   the    P.S., 
which  has  been  located  by  one  of  the  methods  previously 
described,  the  deflection  angle  0   (BAL,  Fig.  1,  page  5) 
will  locate  points  on  the  spiral.     0  may  be  taken  from  the 
tables,  or  it  may  be  calculated  from  equation  (9),  0  =  4- 
A  =  i  a  L~.      For  this  calculation,  if  desired,  the  square  of 
L  may  be  taken  from  a  table  of  squares,  the  lower  deci- 
mals dropped,  and  the  multiplication  by  the  simple  factors 
remaining  may  be  made  easily  and  rapidly.     Thus,  when 
a  —  2,  to  determine  0  for  a  point  234  ft.    (2.34  stations) 
from  the  P.S.,  find  the  square  of  234  (54756),  change  the 


TRANSIT   ON    SPIRAL  39 

decimal  point  so  that  it  will  become  the  square  of  2.34 
(5.48),  and  0  =  I  a  U  =  j  x  2  x  5.48  =  1°  49'.  If  the  re- 
sult is  wanted  in  minutes,  since  J  x  60  =  10,  use  10  instead 
of  J.  The  slide  rule  may  be  used  with  advantage. 

For  a  tabulated  spiral,  the  spiral  deflection  angles  may 
be  taken  from  the  table.  Thus,  for  a  =  1 J,  by  Table  V  0 
for  110  ft.  is  15'.  For  114  ft.  interpolate  proportionally 
between  the  tabulated  value  of  0  for  110  and  that  for 
120  ft.  giving  14J'. 

If  it  is  not  desired  that  the  even  stations  be  located, 
the  spiral  may  be  located  by  50-ft.  chords,  or  chords  of 
other  length,  directly  from  the  P.S.  and  the  labor  of  cal- 
culation will  be  reduced. 

69.  Transit  on  spiral.       With  the  transit  on  an  in- 
termediate point  on  the  spiral,  the  tangent  to  the  spiral 
at  this  point  may  be  obtained  by  turning  off   from  the 
chord  to  the  P.S.  as  a  back-sight  the  angle  A — 0  (ARF, 
Fig.  2,  page  9),  where  A  is  the  spiral  intersection  angle 
and  0  is  the  spiral  deflection  angle  at  the  P.S.   for  the 
given  transit  point  (R).     Except  for  extreme  lengths  this 
is  equal  to  20.     Thus,  by  Table  I    (a  =  J)    for  a  transit 
point  400  ft.   from  the  P.S.,  the  required  angle  is  4°— 
1°  20'=  2°  40'  (or  2  (1°  20')  ). 

70.  For  deflection  angles  from  an  intermediate  transit 
point  on  ordinary  circular  curves,  three  methods  are  in 
use  among  engineers : 

(a)  The  measurement  and  record  of  the  angle  between 
the  tangent  to   the   curve   at   the   transit  point   and   the 
chord  to  the  point  to  be  located. 

(b)  The  use  of  the  angle  between  the  chord  connect- 
ing the  transit  point  to  the  P.C.,  and  the  chord  to  the  point 
to  be  located. 


40  LOCATION    BY    TRANSIT    ANGLES 

(c)  The  use  of  the  angle  between  a  line  through  the 
transit  point  parallel  to  the  initial  tangent  and  the  chord 
to  the  point  to  be  located.  Three  corresponding  methods 
may  be  used  with  the  spiral  and  will  be  treated  separately. 

71.  Intermediate  deflection  angles,  (a)  From  tan- 
gent. By  equation  (10),  the  angle  between  the  tangent 
at  the  transit  point  and  any  chord  (as  CBH,  Fig.  4,  page 
21)  is  <b  =  laL'  (L  —  L')  db  ja(L  —  L')2.  See  also  pages 
10  and  20.  This  method  then  involves  the  following 
steps:  With  transit  at  B  (Fig.  4,  page  21)  set  vernier 
at  A'  —  ff,  or  20'  (these  being  the  angles  for  transit  at  the 
P.S.  for  the  point  B),  and  back-sight  on  the  P.S.  so  that 
the  zero  reading  will  give  the  tangent  BH.  To  locate 
any  point  C  find  the  sum  of  (1)  one  half  of  the  product 
of  the  degree-of-curve  at  the  transit  point  B  by  the  dis- 
tance in  stations  from  the  transit  point  to  C  and  (2)  the 
spiral  deflection  angle  0  for  the  same  distance.  For  D, 
find  the  difference  of  these  quantities.  Thus,  for  a=l, 
with  the  transit  300  ft.  from  the  P.S.,  £>'  the  degree-of- 
curve  at  B  is  3°.  By  Table  IV,  ff  (I-  =  3)  is  1°  30',  and 
ABG  is  3°,  giving  the  position  of  the  tangent  at  B.  For 
C  100  ft.  from  B,.  add  Jx3xl  =  l°30'  and  Jxlxl2^ 
10',  giving  1°  40'  for  CBH.  For  D  100  ft.  from  B,  DBG 


72.  (b)  From  chord  to  the  P.  S.  This  is  the  meth- 
od generally  to  be  recommended.  By  equation  (11)  the 
angle  between  the  chord  from  transit  point  to  P.S.  and 
any  chord  (as  CBE,  Fig.  4,  page  21)  is  0  +  JZ7L,  0 
being  the  spiral  deflection  angle  from  the  initial  tangent 
for  the  point  to  be  located,  D'  the  degree-of-curve  at  the 
transit  point,  and  L  the  distance  in  stations  from  P.S.  to 
the  point  to  be  located.  (See  also  pages  10  and  21.) 
This  method  involves  the  following  steps:  With  transit 
at  B  and  vernier  reading  zero,  back-sight  on  the  P.S. 


TRANSIT   AT   P.C.C.  41 

To  locate  C  turn  off  an  angle  equal  to  the  sum  of  (1) 
the  spiral  deflection  angle  0  for  a  distance  equal  to  the 
distance  from  C  to  the  P.S.  and  (2)  one  sixth  of  the 
product  of  the  degree-of-curve  at  the  transit  point  and 
L  for  the  point  C.  Thus  for  a  =  1,  with  the  transit  300 
ft.  from  the  P.S.,  D'  at  B  is  3°.  For  C  100  ft.  from  B 
and  400  ft.  from  the  P.S.,  add  J  x  1  x  42=:20  40'  (which  is 
the  spiral  deflection  angle  0  for  400  ft.)  and  J  x  3  x  4  =  2° 
(which  is  J  D"  L)  giving  4°  40'  for  CBE.  For  D  100  ft. 
from  B,  DBA  =  40'  + 1°  =  1°  40'. 

To  facilitate  the  calculation  the  transit  point  may  be 
chosen  at  a  point  where  the  spiral  has  an  even  degree-of- 
curve,  as  in  the  above  example,  but  this  is  not  essential. 
It  may  be  seen  that  TV  D'  gives  the  minutes  per  foot  in 
J-2XL. 

73.  (c)    Angles  with   initial   tangent.     The  use  of 

angles  with  the  line  parallel  to  the  initial  tangent  (BK, 
Fig.  4)  is  the  same  as  (b)  except  that  ff,  the  spiral  deflec- 
ion  angle  to  the  transit  point,  must  be  added  to  all  angles. 
Otherwise  the  method  is  the  same  as  (b).  Use  equation 
(12). 

74.  Transit  at  P.  C.  C.     With    the    transit    at    the    P. 
C.C.,  the  tangent  to  the  curve  may  be  found  by  turning 
off  from  the  chord  to  the  P.S.  an  angle  A± —  Ov  these  being 
the  angles  for  the  full  spiral.    Within  ordinary  limits  this 
angle  equals  20A.     The  main  circular  curve  may  be  run 
as  usual. 

In  case  the  P.S.  can  not  be  seen  from  the  P.C.C.,  the 
chord  to  the  P.S.  may  be  located  by  turning  off  from 
the  chord  to  an  intermediate  point  on  the  spiral  an  angle 
A!  —  #! —  <E>  (ACB,  Fig.  4,  page  21)  where  <£  is  the  angle 
between  the  chord  and  the  tangent  at  P.C.C.  (BCF). 
(See  page  10.) 

To  locate  the  chord  from  P.C.C.  to  P.C.  (not  shown  in 


42  LOCATION  BY  TRANSIT  ANGLES 

any  diagram),  deflect  from  the  chord  to  the  P.S.  the  angle 
J  At  —  Or  To  locate  chord  to  P.C.  from  a  chord  to  an 
intermediate  point  on  spiral,  deflect  from  chord  to  the 
intermediate  point  the  angle  J  A± —  <£>.  With  the  data 
already  at  hand,  it  may  be  easier  to  calculate  this  angle 
as  0  +  J  D*  L  —  i  AX+  0V  remembering  that  0  and  L  refer 
to  the  intermediate  point  and  D',  A1?  and  0±  to  the  P.C.C. 

75.  For  the  circular  curve  some  engineers  prefer  to 
measure  the   deflection   angles   from  the   tangent   at  the 
P.C.C.,  and  others  prefer  to  measure  from  the  chord  from 
P.C.C.    to    P.C.    and    thus    maintain    the    same    notes    as 
though  the  spiral  had  not  been  used.     By  the  use  of  the 
angles  discussed  in  preceding  paragraphs,  either  method 
may  be  used. 

76.  To  run  from  the  P.C.C.  toward  the  P.S.     Two 

methods  may  be  used,  (a)  using  L  as  the  distance  from 
P.C.C.  and  deflecting  from  the  tangent,  and  (b)  using  L 
as  measured  from  the  P.S.  and  deflecting  from  chord  to 
P.S. 

77.  (a)  Angles  from  tangent.    -Using  the  distance  L 
as  measured  from  the  P.C.C.,  deflect  from  the  tangent  to 
the  curve  an  angle  equal  to  the  difference  of   (1)    one 
half  of  the  product  of  Dt  (degree-of-curve  at  P.C.C.)  and 
distance  L  to  point  (which  is  the  same  as  the  deflection 
angle  for  D±°  circular  curve)    and   (2)    spiral  deflection 
angle  0  for  distance  L,   (J  a  L2).     This  is  the  same  as 
method  (a)  of  "Transit  on  Spiral."    The  method  depends 
upon  the  principle  that  the  spiral  deflects  from  the  oscu- 
lating curve  at  the  P.C.C.  at  the  same  rate  that  it  deflects 
from  the  initial  tangent  at  P.S. 

As  an  example  take  400  ft.  of  spiral  connecting  with  a 
4°  curve  (0  =  1).  Measure  L  from  P.C.C.  For  a  point 
150  ft.  from  P.C.C.  (L  =  1.5)3  take  the  difference  between 


ANGLES   FROM   CHORD 


Jx4xl.5  =  3°  and  22J'  (spiral  deflection  angle  for  150 
ft.)  which  is  2°  37  J'.  This  angle  is  to  be  deflected  from 
the  tangent  at  P.C.C.  By  the  same  method  the  angle  to 
locate  the  P.S.  is  £  x  4  x  4  =  8°  minus  2°  40',  or  5°  20',  the 
result  found  by  the  usual  method. 

78.  (b)  Angles  from  chord  to  P.  S.  Using  the  dis- 
tance L  as  measured  from  the  P.S.,  deflect  from  the  chord 
to  the  P.S.  an  angle  equal  to  the  sum  of  (1)  the  spiral 
deflection  angle  0  for  distance  L  from  the  P.S.  (J  a  U) 
and  (2)  one  sixth  of  the  product  of  D1  (degree-of- 
curve  at  the  P.C.C.)  and  L,  (j  D^  L).  This  is  the  same 
as  (b)  of  "Transit  on  Sprial." 

Using  the  example  cited  in  the  preceding  method,  150 
ft.,  from  P.C.C.  will  be  250  ft.  from  P.S.,  and  L=r2.5. 
0  =  jxlx  (2.5)a=l°2i'.  jJD1L  =  jx4x2.5  =  l°40'  The 
sum  of  these  is  2°  42  J',  the  angle  to  be  deflected  from  the 
chord  to  P.S.  By  the  same  method,  for  P.S.  L  is  0,  and 
the  deflection  angle  proves  to  be  0,  as  it  should  be. 


5TA. 

POINTS 

0 

iD'L 

+29  o 

PCC.  6° 

5-°20' 

J-37'     f 
L*°57l  \ 

2ef  Vernier  at  <p  =•  •+°3&'J 
back-s/4/)t  01  ff.antf  O' 

20 

4e  55"' 

jF3/r~  I 

[7*56'1    • 

fead"i<?  g/ses  fa^^enf. 

f 
+  5O 

7%?6' 

F75, 

(6"  POj 

/5           ° 

o'=*  5:4  ^ 

2-27' 

5ef  Vern/er  af  O?t>ack-5/g/)t 
on  PS..  and  furn  off  angte 

+50 

/'37' 

in  brackets 

/8 

O*5  d' 

/7 

O°/O 

Curve  to  ft/aftf 

/6  -r29<* 

S*S 

0*0' 

a=2,   *-,=  *,  A=/6°. 

FIELD  NOTES 

79.  Transit  notes.  For  a  spiral  with  a  =  2  connect- 
ing with  an  8°  curve,  L  =  4,  and  if  the  P.S.  has  been  found 
to  be  at  16  +  29,  the  notes  may  be  made  as  follows,  using 
method  (b)  for  the  transit  on  the  spiral.  At  Sta.  19  the 


44  APPLICATION  TO  EXISTING  CURVES 

deflection  angle  from  the  chord  to  the  P.S.,  as  a  back- 
sight, is  the  sum  of  those  given  in  third  and  fourth  col- 
umns, and  it  is  here  inclosed  in  brackets. 


APPLICATION  TO  EXISTING  CURVES 

80.  When  a  road  has  been  constructed  without  transi- 
tion   curves,    the   ordinary    application   of   the    preceding 
principles  will  require  a  new  line  to  be  built  inside  the 
old  curve,  and  the  cost  of  construction  may  be  consider- 
able.    To  retain  as  far  as  possible  the  old  roadbed,  three 
methods  are  applicable: 

(0)  To  replace  the  old  curve  with  a  new  and  sharper 
curve  located  so  as  not  to  vary  far  from  the  old  align- 
ment. 

(0)  To  replace  a  part  of  the  existing  curve  with  a 
curve  of  slightly  smaller  radius,  compounding  with  the 
old  curve. 

(c)  To  make  a  new  alignment  for  the  main  part  of 
the  curve  close  to  the  old  and  replace  a  part  of  this  with 
a  curve  of  smaller  radius. 

81.  To  replace  the  entire  curve.    First  method. — In 

Fig.  6,  the  dotted  line  TNH  is  the  old  curve,  T  being  its 
P.C.  It  is  desired  to  throw  the  line  out  at  H,  the  middle 
point  of  the  curve,  a  distance  of  HK  =  />,  and  replace  the 
curve  by  a  sharper  curve  whose  P.C.  will  be  at  D,  thus 
permitting  the  spiral  AEL  to  be  inserted.  P  is  the  inter- 
section of  tangents,  which  comes  outside  the  diagram. 
Let  R^  be  the  radius  of  the  eld  curve  and  R  of  the  new. 
HP  —  KP  =  p,  or 

R!  exsec  J  /  —  (R  +  o)  exsec  J  /  —  o  =  p. 


TO   REPLACE  THE   ENTIRE   CURVE 


Hence 


R  —  R  =  o  + 


O  -t-/>  O  +  p 

exsec  J7  ~~  vers  4  7 
vers  i  7  —  o 


(24) 


—  cos  4  / 

=  (^  —  7?  —  o)  exsec  4  7—  o..(25) 

Also  AT  =  AP  —  TP=t—(Rl^R  —  o)  tan  4  7 

=  ;_(0  +  />)  Cot  i  7 (26) 

by  which  the  P.S.  (A)  may  be  located;  or  if  T  is  not 
known,  the  tangent  distance  AP  may  be  calculated  and  A 
located. 


T        B 


M 


OM 

i 


'H\\ 


FIG.  6. 

82.  Values  of  p  from  zero  to  Jo  may  be  used.  If 
the  new  curve  comes  inside  the  old  at  the  center,  p  must 
be  used  as  negative  and  its  sign  in  the  formula  must  be 
changed.  It  must  be  borne  in  mind  that  the  o  used  in  the 
above  formula  must  be  the  o  of  the  new  curve.  As  this 
will  not  be  known,  first  use  the  value  of  o  for  the  old 


46  APPLICATION  TO  EXISTING  CURVES 

curve  in  (22),  select  a  radius  and  degree  of  new  curve 
near  the  resulting  value,  and  then  determine  p  and  AT 
with  the  o  for  the  new  curve. 

83.  As  an  example  take  7  =  60°,  D  =  6°,  a  =  2.    Then 
o  for  a  6°  curve  is  3.93.    Take  1.0  ft.  as  a  trial  value  of  p. 
By  equation   (24)   the  radius  of  the  new  'curve  will  be 
approximately  35.8  ft.  shorter  than  the  old  and  by  consult- 
ing a  table  of  radii  of  curves  it  will  be  seen  that  a  6°  14' 

curve  may  be  used.        - — -  =3.117;  there  will  be  311.7  ft. 
a 

of  spiral  at  the  end..  The  o  for  a  6°  14'  curve  will  be  4.4 
ft.  and  the  resulting  p  is  found  by  (25)  to  be  0.5  ft.  There 
will  be  9°  43'  in  each  of  the  spirals  and  40°  34'  in  the 
remaining  circular  curve.  The  P.S.  may  be  located  by 
measuring  the  tangent-distance  T,  or  the  middle  point 
K  of  the  curve  may  be  located  by  means  of  the  external 
distance,  E. 

84.  Second    method.         The   method  just   described 
may  be  modified  to  use  measurements  along  the  external 
secant  as  follows,  using  Fig.  6  as  before :     Intersect  tan- 
gents at  P.     (Intersection  outside  of  diagram.)     Measure 
PK  along  external  secant  to  the  point  K  where  it  is  de- 
sired to  have  middle  point  of  new  curve  come.    By  equa- 
tion (20)    (page  15)  calculate  the  radius  and  the  degree- 
of-curve  which  will  give  PK  as  the  external-distance  E 
of  a  spiraled  curve.    It  will  be  necessary  to  use  the  value 
of  o  for  a  degree-of-curve  equal  to  that  of  the  original 
curve,  since  the  degree  of  the  new  curve  is  not  yet  known. 
Next  select  a  curve  whose  degree  will  give  a  radius  close 
to  that  found  by  the  above  calculation.    For  this  D,  com- 
pute 0,  and  also  PK.    As  the  real  o  was  not  knowrn  in  the 
first  calculation  and  the  new  curve  will  not  have  exactly 
the  R  found,  the  point  K  as  now  located  may  not  coincide 
with  that  first  chosen.    Having  located  K  anew,  the  curve 


TO  REPLACE  A  PART  OF  THE  CURVE  47 

may  be  run  in  from  K  with  back-sight  on  P,  or  the  tan- 
gent-distances* may  be  measured  to  locate  P.S. 

Instead  of  using  equation  (20),  PK  may  be  found  by 
adding  o  sec  J  /  to  the  external-distance  for  7°  of  circu- 
lar curve  without  spiral.  Likewise  in  finding  the  desired 
D,  subtract  o  sec  J  /  from  the  measured  distance  PK,  and 
use  the  remainder  as  the  external-distance  for  an  un- 
spiraled  circular  curve.  By  this  means  a  table  of  external- 
distances  for  a  1°  curve  may  be  utilized  and  the  calcula- 
tions shortened. 

85.  This   method   is   applicable   on   short   curves    and 
\vhere  the  ground  will  permit  of  easy  and  accurate  meas- 
urement of  the  external-distance. 

Take  the  same  example  as  before.  Consider  that  the 
measured  PK  is  146.7  ft.  Using  0  =  3.93,  o  sec  J  7  =  4.5 
ft.  The  circular  curve  whose  external-distance  is  146.7  — 
4.5  =  142.2  ft.  lies  between  6°  14'  and  6°  13'.  Choosing  a 
6°  14'  curve  and  recalculating,  the  external-distance  for  a 
simple  curve  is  found  to  be  142.2  and  o  sec  J  /  5.1,  mak- 
ing PK  147.3  ft.  After  K  is  located  the  curve  may  be 
run  in. 

86.  (b)     To  replace  a  part  of  the  curve.    -In  Fig.  7, 
B  is  the  P.C.  of  the  old  curve  whose  degree  is  D0.    It  is 
desired  to  go  back  on  this  curve  a  distance  BD  and  there 
compound  with  a  curve  of  somewhat  sharper  curvature, 
Dlf  which  if  run  to  a  point  E  where  its  tangent  is  parallel 
to  the  original  tangent  shall  be  at  a  distance  EF  =  o  from 
it.     The  tangent  and  D±  curve  may  then  be  connected  by 
a  spiral  having  this  o.    It  is  required  to  locate  D  and  the 
P.S.  and  P.C.C.  so  that  a  selected  curve,  D19  will  give  a 
calculated  or  assumed  distance  EF  as  o. 

Let  R0  be  the  radius  of  the  D0  curve  and  R^  that  of  the 
D^  curve,  and  /±  the  angle  to  be  replaced. 
o  =  EF  ==  FH  —  EH  ==  (#0  —  J?J  vers  /,. 


48 


APPLICATION  TO  EXISTING  CURVES 


vers  7,  —  - 


.  (27) 


Having  Ilt  back  up  on  the  curve  to  D,  run  the  D^  curve 
to  G,  the  P.C.C.  of  spiral,  and  locate  the  spiral. 
The  P.S.  may'  be  located  from  B  by 
AB  =  t—(R9  —  R1  —  o)    tan/, 

=  t—(R0  —  R1)   sin  7,  ...................  (28) 


-Or /ft I? a/  Curve 


FIG.  7. 

87.  Thus,  consider  that  a  part  of  a  4°  curve  is  to  be 
replaced  with  a  4°30'  curve,  and  that  a  =  1.  o  =  6.62.  By 
equation  (27),  71=16°  35'.  Take  out  BD  =  414.6  ft.  of  4° 
curve  and  locate  D.  16°  35'  of  4°  30'  curve  requires  368.5 
ft.  The  half  length  of  the  spiral  is  225  ft.  The  P.C.C.  is 
then  found  by  running  from  D  368.5  —  225  =  143.5  ft.  of 
4°  30'  curve  to  the  P.P.C.,  G.  Likewise  by  (28)  A.B.  = 
224.8  —  45.4  =  179.4  ft. 


TO  RE-ALIGN    AND   COMPOUND 


49 


88.  The  limiting  values  of  Di  will  be  on  the  one  hand 
|  D0  and  on  the  other  a  value  which  will  make  BD  one 
half  of  the  length  of  the  original  curve.     Ordinarily,  Dt 
should  not  be  one  fifth  more  than  D0;  better  less  than  one 
tenth  more  on  sharp  curves. 

89.  It  may  be  convenient  to  calculate  a  standard  set 
of  values  for  the  curves  on  a  road,     The  following  gives 
a  few  such  values. 


DQ 

a 

A 

^o—^i 

0 

/, 

AB 

GD 

2° 

i 

2°15' 

318.3 

3.31 

8°16' 

179.2 

142.4 

2° 

i' 

2°30/ 

572.9 

4.53 

7°12' 

178.1 

38.0 

3° 

1 

3°  30' 

272.8 

3.12 

8°40' 

133.8 

72.6 

4° 

1 

5° 

286.4 

9.07 

14°27' 

178.1 

39.0 

5° 

1 

6° 

190.9 

15.65 

23°21' 

223.4 

89.2 

5° 

2 

6°  ' 

190.9 

3.91 

ll°38r 

111.4 

43.9 

90.  (c)  To  re-align  and  compound.  When  the 
middle  portion  of  the  curve  is  in  fair  alignment  and  it  is 
desired  not  to  disturb  it,  or  when  it  seems  best  to  re-locate 
the  central  part  of  the  curve,  a  method  by  taking  up 
points  on  the  old  track  and  not  running  the  principal  tan- 
gents to  an  intersection,  may  be  used.  See  Fig.  8.  Select 
M,  N,  and  O  on  the  curve  on  the  portion  not  to  be  dis- 
turbed. Set  transit  at  M,  measure  the  distances  MN  and 
NO,  and  by  the  usual  methods  for  circular  curves  deter- 
mine the  degree  of  curve,  D0,  which  will  fit  this  middle 
portion.  Or  select  points  that  will  locate  the  curve  in  a 
desirable  position,  and  determine  DQ.  The  selection  of 
points  in  this  way  will  probably  not  give  a  curve  whose 
tangent  coincides  with  the  track  tangent.  When  this 
curve  is  run  back  until  its  tangent  is  parallel  to  AH  at 
B,  the  distance  from  the  track  tangent  will  be  called  m. 
From  M,  intersect  with  tangent  at  H  and  measure  /. 
Determine  m  by  running  out  MDB  and  measuring  the 


APPLICATION  TO  EXISTING  CURVES 


offset  or  by  calculation  from  m=HG  sin  /  and  HG=HM 
—  GM,  remembering  that  GM  is  the  tangent-distance  for 
/°  of  D0  curve.  Let  ED  be  the  new  Dt  curve  which  must 
be  run  in  from  D  so  that  EF  shall  be  the  o  for  the  Dt 
curve.  Call  the  radius  of  the  D0  curve  R0  and  that  of  the 
D±  curve  Rr  The  D^  curve  will  have  /±  of  central  angle. 
Then 

EJH=o  —  m=(R0  —  R1)  vers  /t. 
o  —  m 


vers  /.  =  - 


.(29) 


If  o  is  less  than  m,  then  Rt  must  be  greater  than  Ra. 


FIG.  8. 

If  K  comes  outside  of  AH,  m  must  be  added  to  o.    Care 
must  be  taken  that  M  is  far  enough  back  on  the  curve. 

91.  To  locate  the  curve,  run  /  —  /.of  D0  curve  from 
M  to  D.  Run  DE  to  locate  the  P.C.,  or  run  such  part  of 
this  D1  curve  as  will  give  the  P.C.C.  for  the  spiral. 


METHODS  OF  TRACK   MEN  51 

The  P.S.   (A)  may  be  located  as  follows: 

AH  =  *  +  BG—  BK  —  HG  cos  / (30) 

BG  =  GM,  the  tangent-distance  for  7°  of  D0  curve,  BK  = 
(^0— RJ  sin  7,,  and  HG  cos  7  is  also  m  cot  7.  The  P.S. 
may  also  be  located  by  offsetting  o  from  E  to  F  and 
measuring  t  to  A. 

92.  For  example,  if  D0  has  been  found  to  be  4°  and  7 
at  H  20°  and  m  1.2  ft.,  select  D^  =  4°  30'  and  a  =  I.    Then 
c  =  6.62.     From  equation  (29)  7,=  15°     Then  calculating 
the  length  of  the  curves  from  the  angles  7  and  Ilf  as  MB 
is  500  ft.  and  DB  375  ft,  MD  =  125  ft.     DE  =  333.3  ft. 
since  there  is  to  be  15°  of  4°  30'  curve.     The  P.C.C.  for 
spiral  is  333.3  —  225  =  108.3  ft.  from  D  toward  E,  since 
the  half  length  of  the  spiral  is  225  ft.    AH  is  224.8  +  253.6 
—  41.1  —  3.3  =  434.0   ft.     The  spiral  may  be  run   in  by 
usual  methods. 

93.  The   limiting  values   of  D1   are  similar  to  those 
given   in   the   preceding  method.     Generally  D1  may   be 
from  one  tenth  to  one  fourth  more  than  D0,  depending 
upon  the  amount  of  the  curve  and  its  degree. 

94.  When  the  new  7?±  curve  is  so  much  sharper  that 
it  is  desired  to  connect  it  with  the  old  by  a  spiral,  the 
following  method  is  applicable.     Call  ot  the  offset  to  tan- 
gent, and  o0  the  offset  between  the  two  curves,  the  latter 
to  be  found  as  for  compound  curves.     Then  by  a  method 
similar  to  the  foregoing, 


(31) 


95.  Methods  of  track  men.  When  curves  are  left 
without  transition  curves,  many  track  men  "ease"  the 
curve  by  throwing  the  P.C.  inward  a  short  distance  and 
gradually  approaching  the  tangent  a  few  rail  lengths 


52  APPLICATION  TO  EXISTING  CURVES 

away,  while  the  main  curve  is  reached  finally  by  sharpen- 
ing the  curve  for  a  short  distance. 

96.  Another  simple  method  for  track  which  is  aligned 
to  a  circular  curve,  consists  in  utilizing  one  of  the  proper- 
ties of  the  transition  spiral.     In  Fig.  1,  page  5,  let  ABK 
be  the  original   track  line,   B   being  the   P.C.      Select   a 
length  of  spiral  and  calculate  o,  or  select  o  and  calculate 
the  length,  by  a  preceding  method.     At  a  distance  from 
B  equal  to  half  the  length  of  spiral   (point  of  the  curve 
opposite  L)  throw  the  track  inward  to  L  a  distance  equal 
to  o.    At  B,  the  old  P.C.,  throw  the  track  to  E,  a  distance 
half  as  great.     Measure  back  from  B  half  the  length  of 
the  spiral  to  A  for  the  beginning  of  the  easement.     Be- 
tween A  and  L,  line  the  track  by  eye,  or  calculate  offsets 
from  Table  IX.     The  remainder  of  the  main  curve  must 
then  be  thrown  in  the  same  distance  as  at  L. 

97.  On  long  curves  the  latter  work  would  be  objec- 
tionable.    It  may  be  avoided  by  using  a  spiral  running 
up  to  a  curve  whose  degree-of-curve  is  one  third  greater 
than  that  of  the  main  curve  and  compounding  directly 
with  the  main  curve.     To  do  this,  first  select  length  of 
spiral    for   a  curve   one  third  sharper  than   the   circular 
curve  which  call  L.      See  Fig.  9.    Call  the  circular  curve 
D0  and  the  curvature  of  the  end  of  the  spiral  Dr  Measure 
back  from  the  old  P.C.  on  tangent  a  distance  J  L,  which 
will  locate  the  P.S.    Measure  forward  on  the  curve  from 
the  P.C.  a  distance  J  L  to  locate  the  middle  of  the  spiral, 
and  offset  from  prolongation  of  tangent  a  distance  equal 
to  J  o,  or  J  o  from  the  circular  curve.     Measure  also 
along  the  curve  from  the  P.C.  a  distance  |  L  to  the  P.C.C. 
where   the  track  will   not  be  changed.     The  spiral  will 
pass  the  old  P.C.  at  -£^0  from  it,  and  at  a  point  J  L  from 
the  P.C.C.  will  be   J0  distant  from  the   circular  curve. 


METHODS   OF  TRACK   MEN 


53 


The  spiral  is  one  and  one  half  times  as  long  as  the  cir- 
cular curve  replaced. 

The  o  used  must  be  that  for  the  full  spiral  and  for  the 
sharper  curve,  |r  DQ,  and  the  true  position  of  the  cir- 
cular curve  should  be  known.  As  the  last  fourth  of  this 
spiral  is  sharper  than  the  main  curve,  the  elevation  of 
the  other  rail  up  to  the  P.C.C.  must  be  greater  than  that 
on  the  main  curve,  gradually  reducing  beyond  to  the 
regular  amount. 


J_ 


,0/JP.c.      ,J. 


P.  5. 


2.  o 


U"' 


FIG.  9. 


98.  Thus,  for  a  3°  curve  using  a  =  l,  the  degree  at 
the  end  of  the  spiral  will  be  3  x  |^  =4,  and  the  length  of 
spiral  required  is  400  ft.,  0  =  4.65.  The  P.S.  will  be  133.3 
ft.  back  of  the  P.C.,  the  middle  of  spiral  66.7  ft.  ahead 
of  P.C.  and  the  P.C.C.  266.T  ft.  ahead  of  P.C.  At  the 
P.C.  the  track  must  be  thrown  in  0.69  ft.,  at  the  middle 
point  2.32  ft.  from  tangent  (1.16  ft.  from  curve),  and  at 
the  third  quarter  point  .58  ft.  from  curve,  while  at  the 


54  COMPOUND  CURVES 

P.C.C.  there  will  be  no  change.  Between  these  points  the 
track  may  be  aligned  by  eye,  or  ordinates  may  be  calcu- 
lated by  Table  XII. 

However,  while  such  methods  are  easements,  they  are 
at  best  makeshifts  and  should  give  place  to  better 
methods. 


COMPOUND  CURVES. 

99.  The  spiral  may  be  used  to  connect  curves  of  dif- 
ferent radii,  choosing  that  part  of  the  spiral  having  curv- 
ature intermediate  between  the  degrees  of  the  two 
curves,  thus,  connect  a  3°  and  an  8°  curve  by  omitting 
the  spiral  up  to  D  =  3°  and  continuing  until  D  =  8°.  In 
Fig.  10,  DKM  is  a  Di  curve,  and  LNP  a  D2  curve,  the 
two  curves  having  parallel  tangents  at  M  and  N.  D2  is 
greater  than  Dr  Call  the  distance  MN  o.  It  is  desired 
to  connect  the  two  curves  by  a  spiral  shown  by  the  full 
line  KP.  The  degree  of  curve  of  the  spiral  at  K  must  be 
DI  and  at  P?  D2.  Consider  the  spiral  to  be  run  backward 
from  K  to  a  tangent  at  A.  Then  the  spiral  from  K  to 
P  is  the  portion  of  the  regular  spiral  from  where  its 
degree  is  D^  to  the  point  where  it  is  D2.  Since  the  spiral 
diverges  from  the  osculating  circle  at  the  same  rate  as 
from  the  tangent  at  the  P.S.,  PN  =  MK  and  the  spiral 
bisects  MN.  MN,  or  o,  is  the  offset  for  a  spiral  for  a 
curve  whose  degree  is  D2 —  Dr  Hence,  find  o  for  a  Dz 
—  D1  curve,  and  make  the  offset  at  MN.  Measure  MK 

and  NP  each  equal  to  J      2~    *    thus  locating  the  P.C.C. 

of  each  curve  K  and  P.  Run  in  the  spiral  from  K  or  P 
by  the  method  for  point  on  spiral  heretofore  described, 
AK  being  omitted.  The  angle  between  tangents  at  K  and 


TO  INSERT  IN  OLD  TRACK 


55 


P  is  A  for  a  D2  spiral  minus  A  for  D1  spiral,  and  may  also 
be  expressed  as  £  (D1  +  D2)  limes  KP  in  stations.  Thus, 
with  a  =  2,  to  connect  a  3°  and  an  8°  curve  o  =2.27,  the 
value  for  a  5°  spiral.  The  portion  of  the  spiral  used  will 
be  250  ft.  long.  K  is  125  ft.  from  M.  and  N  is  125  ft. 
from  P.  If  greater  accuracy  is  required,  the  t  COR.  for 


FIG.  10. 

this  length  should  be  subtracted  from  125.  The  angle 
between  tangents  at  K  and  P  is  £  x  2.50  (3°  +  8°)  = 
13°  45'. 

The  spiral  may  also  be  used  to  connect  two  curves 
having  a  given  offset  between  them. 

100.  To  insert  in  old  track.  It  may  be  desired  to 
insert  a  spiral  between  the  two  curves  of  an  existing  com- 
pound curve  by  first  replacing  a  part  of  the  sharper  curve 
with  a  curve  of  slightly  smaller  radius. 

In  Fig.  11,  let  AB  be  a  £>±  curve  and  BG  a  D3  curve, 
B  being  the  P.C.C.  and  the  D3  -curve  having  the  smaller 
tadius.  Q  is  the  center  of  the  D1  curve,  not  on  the  cut, 


56  COMPOUND  CURVES 

It  is  desired  to  go  back  on  the  D3  curve  to  a  point  D  and 
there  compound  with  a  Dz  curve  which  shall  be  run  to  a 
point  E  where  its  tangent  shall  have  the  same  direction 
as  the  tangent  to  the  DI  curve  produced  backward  to  F 
has  at  F.  The  radial  distance  EF  corresponds  to  the 
offset  of  the  usual  spiral  and  will  be  called  o.  It  is  de- 
sired to  locate  D  and  F  L'O  that  a  selected  cuve,  D2,  will 
give  a  calculated  or  assumed  distance  EF  as  o. 

The  distance  EF  is  made  up  of  FK  and  KE,  the  first 
being  the  divergence  of  the  D±  curve  from  the  D3  curve  in 


the  distance  BF  and  the  second  the  divergence  of  the  D2 
curve  from  the  Ds  in  the  distance  DE.  Call  the  distance 
BF  Lv  and  DE  L2..  For  the  small  angles  used  these 
divergences  may  be  calculated  accurately  enough  by  the 
approximate  formula  for  tangent  offset,  y==.87  DUy  and 
we  shall  have 

EF  =  .87  (D2  —  D3)  L/+.87  (£>3  — £>,)  L?  =  o,or 
(D2  —  DZ)   L22+(D3  —  Dl)   L^  1.150 (32) 

Since  the  amount  of  D^  curve  in  BF  plus  the  amount 


TO    INSERT    IN    OLD    TRACK  57 

of  D2  curve  in  DE   (total  angle)  must  be  equal  to  the 
amount  of  Ds  curve  taken  out,  we  have 
I>,L,  +  I>1L1  =  .y,(L1  +  L1)   or 

(^.-^.)    L2=(^-^)  /-,...  ...  .....  ..(33) 

Combining  (32)  and  (33)  and  solving, 


V        3  I/  /OK\ 

-  ^     ...........  (35) 


101.  Having  L±  and  L2,  the  points  D,  E  and  F  may 
be  located,  and  the  D2  curve  may  be  run  in  from  D  as 
far  as  necessary.  The  problem  is  then  identical  with 
that  of  putting  a  spiral  between  two  curves  having  an 
offset  o  (EF)  between  their  parallel  tangents. 

By  the  principles  governing  the  placing  of  a  spiral  be- 
tween two  curves,  it  is  seen  that  the  length  of  the  con- 
necting spiral  L'  is  that  of  a  spiral  for  a  curve  of  degree 
equal  to  the  difference  of  degree  of  the  two  connected; 
that  is 


a 
The  offset  is  equal  to  that  for  a  (D2  —  D^)  degree  curve 

from  a  tangent  or 
o  =  .0727  (D2  —  DJ  L''2  =  .0727  aL"  ............  (36) 

Half  of  this  spiral  will  lie  on  one  side  of  the  offset  and 

half  on  the  other,  hence  in  Fig.  12  JL  to  the  right  of  F 

will  give  the  beginning  of  the  spiral,  H,  and  JL  to  the 

left  of  E  will  give  the  end  of  spiral,  I. 

102.  The  method  of  field  work  will  then  be  as  follows  : 
Measure  from  B,  the  P.C.C.,  (Fig.  12)  back  on  the  Dt 
curve  a  distance  BH  =  J  L'  —  L±  to  locate  the  point  of 
spiral  H.  Measure  from  B  on  the  Dz  curve  the  distance 
BD  =  Lt  +  L2  to  D,  the  new  P.C.C.,  run  in  the  D2  curve  to 
I,  DI  being  L2  —  i  I/  .  The  spiral  is  then  to  be  run  in 


58 


COMPOUND  CURVES 


from  H  to  I.    The  dotted  line  in  Fig.  12  shows  the  spiral. 

The  field  work  for  the  spiral  is  simple.  The  spiral  may 
be  run  in  by  offsetting  from  the  D^  curve  HF  (Fig.  12) 
knowing  that  the  offset  from  the  curve  to  the  spiral  is 
the  same  as  that  of  a  spiral  from  the  tangent  using  the 
distance  from  H  as  the  distance  on  the  spiral.  Likewise 
the  remainder  of  the  spiral  may  be  ofTsetted  from  the  D., 
curve  IE  using  distances  from  I  in  the  calculations. 

If  the  field  work  on  the  spiral  is  to  be  done  by  deflec- 


r 


FIG.  12. 

tion  angles,  the  spiral  may  be  run  in  from  H  by  using 
as  deflection  angles  the  sum  of  the  deflection  angle  for 
the  circular  curve  HF  and  the  spiral  deflection  angle 
from  a  tangent  for  the  same  distance ;  or  the  transition 
spiral  may  be  run  backward  from  I  in  a  similar  manner. 
In  either  case  the  work  will  be  no  more  difficult  than  for 
spirals  for  simple  curves. 


TO   INSERT   IN   OLD  TRACK  59 

103.  As  an  example  let  us  consider  that  a  2°  and  an 
8°  curve  are  compounded  at  B.    Consider  that  the  degree 
of  the  new  curve  to  be  run  in  is  8°  30',   and  that  the 
value  of  a  to  be  used  is  2.    Then  D±  =  2,  D3  =  8,  D2  =  8J. 
For  a  spiral  from  2°  to  8°  30',  the  value  of  the  offset  o 
(EF)  is  the  same  as  the  o  for  a  6°  30'  curve  from  a  tan- 
gent.   Hence  0  =  4.99.     By  formula  (34),  Ll==.271,  and 
by  formula  (35),  L2  =  3.255.     Hence  the  point  D  will  be 
back  on  the  £>3  curve  325.5  +  27.1   or  352.6   ft.   from   B. 
The  length  of  the  spiral  to  be  used  will  be 

L'=  8^~2  =  3.25. 

Of  this  162.5  ft.  will  be  to  the  left  of  E  and  162.5  ft.  will 
be  to  the  right  of  F.  Hence  H  and  I,  the  ends  of  the 
spiral,  may  readily  be  located  and  the  spiral  may  be 
run  in. 

104.  By  this  method  the  value  of  a  may  be  chosen 
beforehand,  the  value  of  o  may  be  easily  calculated,  and 
the  preliminary  field  work  is  small.    It  may  be  stated  that 
the  limiting  values  of  D2  will  be,  on  the.  one  hand,  a  value 
so  near  D3  that  the  resulting  L2  will  carry  the  new  point 
of  compound  curve  back  to  the  end  of  the  old  curve,  and 
on  the  other  hand  such  that  the  length  of  the  D2  curve 
shall  be  at  least  equal  to  half  the  length  of  the  transition 
spiral,   a   value   which   may  be   shown   to  be  D2  =  J    (4 

*>,-• D,). 

For  large  angles  the  above  method  is  subject  to  slight 
error. 


60 


MISCELLANEOUS  PROBLEMS 


MISCELLANEOUS    PROBLEMS 

105.  To  change  tangent  between  curves   of  oppo- 
site  direction.     Having   given   two    curves    of    opposite 
direction  connected  by  a  short  tangent,  it  is  required  to 
find  the  position  of  a  line  to  which  both  curves  may  be 
connected  by  spirals.    This  involves  determining  the  angle 
which  must  be  added  to  each  curve  to  get  the  position  of 
the  new  P.C.  of  each  curve  for  spiraling. 

106.  In  Fig.  13  let  AB  be  the  original  tangent  con- 
necting the  two  curves  and  I  its  length   in   feet.     It  is 
required  to  run  the  curve  KA  to  E,  which  will  be  the 


Or/ginal  Tangent   ^         [^L^^ 
C              N                  ^^ 

r 

4=^  

,,-  '      L--  P     \^ 

AI    Lr      ^r              __               ) 
.i^^^--'  >i     A^*v  Tangent  - 

D 

FIG.  13 


P.C.  for  the  spiraled  curve,  and  MB  to  F  for  its  spiraled 
P.C.,  and  also  to  find  the  position  of  the  line  CD,  which 
will  be  the  common  tangent  for  the  two  spirals.  Call  the 


CURVES   OF   OPPOSITE   DIRECTION  61 

angle  APC  a.  It  is  the  same  as  that  of  the  additional 
amount  of  curve  AE  and  BF.  Let  R±  be  the  radius  of 
the  curve  KE  and  R2  that  of  ME,  and  ol  and  o2  be  the 
respective  spiral  offsets  NE  and  OF  for  the  spirals  chosen 
lor  the  two  curves.  AC  +  BD  =  NE  +  EG  +  OF  +  FH. 
Then,  since  AC  +  BD  =  /  sin  a  and  EG  =  ^  vers  a,  etc., 
Z  sin  a=ol  +  o2+  (R^  +  #2)vers  a, (37) 


107.  Since  a  will  be  small,  we  may  substitute,  using  a 
in  degrees,  sin  a  —  .01745  a  and  vers  <*  =  . 000152  «2  which 
are  close  approximations  below  8°.  Transforming, 

oi  +  o2         .000152  (flj  +  R.,)  a2 
a  ~  .01745  I  "*"  .01745  / 

This  quadratic  may  be  solved,  but  usually  the  follow- 
ing approximate  root  gives  sufficiently  close  results : 
o,  +  ot         .000152   (R,+R9)    (i+om 


-  (.01745  /)* 

Having  a  the  lengths  AE  and  BF  may  be  found,  the 
position  of  the  P.C.C.  of  each  curve  found,  and  the  new 
tangent  located  by  offsetting  EN  and  FO,  or  by  offsetting 
AC  (equal  to  o1  +  R1  versa)  and  BD.  For  very  short 
tangents,  spirals  must  be  chosen  short  enough  not  to  over- 
lap on  the  tangent. 

108.  As  an  example  take  a  3°  curve  and  a  4°  curve 
connected  by  600  ft.  of  tangent.  Use  a  =  1.  Then 

oi  =  1.96  and  02  =  4.65.     By  equation  (38) 

a  =  .63  +  .02  =  .65°  =  0°  39'. 

This  result  checks  equation  (37)  very  closely.  0°39' 
gives  21.7  ft.  of  3°  curve  (AE)  and  16.2  ft.  of  4°  curve 
(BF).  There  will  be  300  ft.  of  spiral  for  the  3°  curve 
and  400  ft.  of  spiral  for  the  4°  curve.  The  P.S.  and 
P.C.C.  of  each  spiral  will  be  half  of  the  spiral  length 
from  the  points  E  and  F.  The  P.C.C.  of  one  will  be  (150 
—21.7=128.3)  ft.  back  of  A,  and  of  the  other  (200-— 16.2 


62  MISCELLANEOUS   PROBLEMS 

=183.8)  ft.  back  of  B.  AC  will  be  (1.96  +  .11  =  2.07) 
ft.  and  BD  (4.65  +.09  =4.74)  ft.  The  distance  from  C  to 
P.S.  will  be  150  +  21.7,  and  from  D  to  the  other  P.S,  200 
+  16.2,  neglecting  the  t  correction.  The  spirals  may  then 
be  run  in  as  usual. 

109.  This  solution  may  also  be  applied  to  the  case 
where  a  tangent  thrown  off  from  the  curve  KA  does  not 
strike  the  curve  MB  but  is  parallel  to  this  curve  at  a 
point  opposite  B  distant  m  from  it.    Since  cos  a  is  nearly 
1,  equations  (37)  and  (38)  may  be  modified  by  subtract- 
ing m  from   (o^  +  oj  wherever  it  occurs.     This  modifica- 
tion is  of  convenience  in  revising  old  lines.    The  engineer 
should  make  his  own  diagram. 

110.  To  change  tangent  between  curves  of  same 
direction.       Having  given  two  curves  of  same  direction 
connected  by  a  tangent  it  is  desired  to  find  the  position 
of  a  line  to  which  the  two  curves  may  be  connected  by 
spirals.     As  in  the  preceding  problem  this  involves  deter- 
mining the  change  in  the  angle  of  the  two  curves  and  the 
position  of  the  P.C.  of  each  curve  for  spiraling. 

111.  In  Fig.  14  let  AB  be  the  original  tangent  con- 
necting the  two  curves  and  I  its  length   in   feet.     It  is 
required  to  back  up  on  the  curve  AK  to  E  for  the  P.C. 
for  spiraled  curve  and  to  run  the  curve  MB  to  F  for  its 
spiraled   P.C.,   and  to   find  the   position   of  the   line   CO 
which  will  be  the  common  tangent  for  the  two  spirals. 
Call  the  angle  BAD'  a.     It  is  the  same  as  that  in  AE 
and  BF.     Let  R^  be  the  radius  of  the  curve  KE  and  R2 
that  of  MF,  and  o^  and  o2  be  the  respective  spiral  offsets 
NE  and  OF  for  the  spirals  chosen  for  the  two  curves. 

BD  —  AC  =  OF+  FH  —  NE  —  EG. 
Then,   since  BD — AC  or  BD'  equals  /  sin  a  and  EG 
equals  Rl  vers  a,  etc. 


CURVES  OF  SAME  DIRECTION 

sin  a  =  o2  —  ol —  (Rt —  R.J)  vers  a. 

A/ew  Tangent  -^ 

N  C \  O 


63 
.(39) 


FIG.  14. 

112.  Since  a  will  be  even  smaller  than  in  the  preced- 
ing problem,  we  may  substitute  sin  a  =  .01745a  and  versa 
=  .000152 a2,  using  a  in  degrees.  Transforming, 

Q.-OI          .000152  (R-R2)     2 
.01745  I  .01745  I 

As  in  the  preceding  problem,  the  approximate  solution 
of  this  quadratic  may  be  used. 

,— A       .000152  (R—R9)  (o 


, 

(in degrees)  =-~  ^-^  ~ 

.01745  I 


(.01745  O3" 
The  last  term  here  is  very  small. 

Having  a  the  lengths  AE  and  BF  may  be  found,  the 
position  of  the  P.C.C.  of  each  curve  found,  and  the  new 
tangent  located  by  offsetting  EN  and  FO,  or  bv  offsetting 
AC  (equal  to  0^  +  ^  vers  a)  and  BD  (equal  to  o.2  +  R: 
vers  a.  The  length  of  spiral  must  not  be  so  great  that 
the  spirals  will  overlap  on  short  tangents. 


64  UNIFORM   CHORD  LENGTH    METHOD 

113.  As  an  example  take  a  3°  curve  and  a  4°  curve 
connected  by  600  ft.  of  tangent.     Use  a  =  1.     Then  OL  = 
1.96  and  02  =  4.65.     By  equation  (40)  a  =.257  —  .006  = 
.251  =  0°15i'. 

0°  15-J'  gives  8.6  ft.  of  3°  curve  ( AE)  and  6.4  ft.  of  4° 
curve  (BF).  There  will  be  300  ft.  of  spiral  for  the  3° 
curve  and  400  ft.  for  the  4°  curve.  The  P.C.C.  of  spiral 
will  then  be  (150  +  8.6  =  158.6)  ft.  back  of  A  and  (200  — 
6.4  =  193.6)  ft.  back  of  B.  AC  will  be  1.98  ft.  and  BD 
4.67  ft.  The  distance  from  C  to  P.S  will  be  150  —  8.6 
and  from  D  to  the  other  P.S.  200  +  6.4,  neglecting  the  t 
correction.  The  spirals  may  then  be  run  in. 

114.  This   solution  may  also  be  applied  to  the  case 
where  a  tangent  thrown  off  from  the  curve  KA  misses 
the  curve  MB  by  a  distance  m  from  B,  the  point  of  par- 
allelism.    In  this  case  equations   (39)   and   (40)   may  be 
modified  by   subtracting  m   from    (o2  —  oj    wherever   it 
occurs.     If  the  second  curve  is  one  of  larger  radius,  it 
will  be  necessary  to  construct  a  new  diagram  and  deter- 
mine the  signs  of  the  terms. 


UNIFORM  CHORD  LENGTH  METHOD 

115.  The  treatment  of  the  spiral  heretofore  given  is 
based  upon  principles  which  permit  the  use  of  any  chord 
length,  either  uniform  or  variable,  throughout  the  length 
of  the  spiral.  Regular  chord  lengths,  like  20  or  25  feet, 
may  be  used,  if  desired  and  the  excess  if  any  used  as  a 
fractional  chord  at  the  beginning  or  the  end  of  the  spiral. 
If  it  is  desired  to  use  chords  of  common  length,  another 
method  known  as  uniform  chord  length  method,  may  be 
derived  by  modifying  the  preceding  formulas.  A  further 
modification  of  this  method  may  be  made  to  allow  the 


UNIFORM   CHORD  LENGTH    METHOD  65 

use  of  fractional  chord  lengths  at  the  beginning  or  the 
end  of  the  spiral,  so  that  it  will  not  be  necessary  to  make 
the  uniform  chord  length  an  aliquot  part  of  the  length 
of  the  spiral.  Thus,  if  the  spiral  is  to  be  203.2  ft.  long, 
ten  20-ft.  chords,  or  eight  25-ft.  chords,  or  thirteen  15-ft. 
chords,  etc.,  may  be  used — the  first  or  last  chord  or  both 
being  fractional. 

116.  The  notation  used  will  be  the  same  as  heretofore 
except  as  noted,  and  the  equations  will  be  numbered  the 
same,  using  the  prime  mark  to  distinguish  them.  Let  c 
be  the  chord  length  used.  This  will  be  expressed  in  hun- 
dreds of  feet,  that  is  in  the  number  of  100-ft.  stations. 


FIG.  15. 

For  a  chord  length  of  20  ft.,  c  =.2 ;  for  one  of  15  ft.,  c  = 
.15,  etc.  Let  n  (an  integer)  represent  the  number  of  full 
chords  from  the  P.S.  to  a  desired  point.  In  Fig.  15,  A 
is  the  P.S.  and  its  n  is  0.  The  n  of  B  is  1,  of  C  2,  etc. 
Let  Oi  =  spiral  deflection  angle  at  P.S.  from  initial  tangent 
for  a  single  full  chord  length  (BAE)  (called  unit  spiral 
deflection  angle),  and  On  for  n  chord  lengths.  For  Dn  =• 


66  UNIFORM   CHORD  LENGTH    METHOD 

3  and  0n  =  DAE.  Similarly,  An,  Dn  and  Ln  are  for  a  point 
n  chord  lengths  from  the  P.S.  For  the  instrument  at 
other  points  than  P.S.,  let  n'  be  the  number  of  chord 
lengths  from  the  P.S.  to  the  chord  point  at  which  the 
instrument  is  located,  reserving  n  still  as  the  number  of 
chord  lengths  from  the  P.S.  to  the  point  to  be  located, 
and  let  3>n  represent  the  deflection  angle  from  tangent  at 
the  instrument  point  to  this  desired  point.  Thus,  if  the 
instrument  is  at  C  two  chord  lengths  from  the  P.S.  nf  = 
2,  and  to  locate  D  three  chord  lengths  from  the  P.S., 
n  =  3  and  <£»w  will  represent  the  deflection  angle  DCK  to 
locate  the  third  chord  point  D. 

117.  The  length  L  =  nc.  By  substitution  in  equations 
(1),  (2),  and  (9)  of  the  spiral,  we  have  for  any  point 
on  the  spiral  distant  L  =  nc  from  the  P.S. 

D  =  a.L  =  a  n  c... (V) 

A  =  i  a  L2  =  J  a  n2  c2 (2') 

For  end  of  first  full  chord  and  for  end  of  n  full  chords, 
respectively, 


(9') 


Also  3>n 

=  1  an'  c  (n  —  «')  c  =b  J  a  (n  —  H') 

=  JW    (n  —  n'):±=(n  —  »')']   #1 
In   formula    (10'),   the   arithmetical   difference   of   the 
numbers   of  the   chord  points   is  taken,   rather  than   the 
algebraic  difference.     If  the  latter  is  used,  the  signs  of. 
operation  should  all  be  plus. 

118.  The  first  step  is  to  calculate  the  value  of  the 
unit  spiral  deflection  angle  0X  by  means  of  equation  (9'), 
using  a  and  c  or  other  terms.  For  a  chord  length  of  20 
ft,  and  a  value  of  a  =  2,  c  =  3  and  ^=J  x  2  x  (2)3  = 


DEFLECTION  ANGLES  67 

0°  0.8'.  If  a  spiral  250  ft.  long  is  to  connect  with  a  4° 
curve  using  25-ft.  chords,  Dn=  4  and  Ln=  2.5,  Ol=  J  + 
4XT16X^=-0°1/.  If  A»  =  9°,  L  =  3andc  =  .2  (20ft), 
ft  =  i  (I)2 x  9°  =  0.8'. 

119.  The  value  of  0±  gives  a  basis  for  computing  the 
deflection  angle  for  other  points;  thus  for  a  point  5  chord 
lengths  from  the  P.S.,  n  =  5  and  the  deflection  angle  by 
equation  (9')  is  25  times  the  value  of  0±.     For  the  instru- 
ment at  the  fifth  chord  point  (w'  =  5),  the  deflection  angle 
from  the  tangent  at  the  instrument  point  to  a  point  8 
chord  lengths  from  the  P.S.  (n  =  S)  is  by  equation  (10')  : 
$nz=:[3x5    (8  — 5)  =±=  (8  — 5)2]   ^  =  54^.     To   locate 
from  the  same  instrument  point  a  point  3  chord  lengths 
from  the  P.S.  (2  from  the  instrument  point),  the  deflec- 
tion angle  is  26  Or 

120.  Table  of  unit  spiral  deflection  angles.     A  table 
giving  0!  for  various  chord  lengths  for  many  of  the  values 
of  a  used  in  field  work  may  be  of  service.     Table  XIII 
gives  spiral  deflection  angles  for  first  chord  length   (unit 
spiral  deflection  angles).     The  angle  is  given  in  minutes. 

,  It  is  well  in  the  calculations  to  express  decimals  as  com- 
mon fractions;  thus,  for  20- ft.  chords  with  a  =  l  use  0, 
=  .5J';  for  16-ft.  chords  with  a  =  l§  use  ^  =  .421'. 

121.  Table  of  coefficients  for  deflection  angles.    The 

values  obtained  from  (9')  and  (10')  may  be  considered 
as  coefficients  of  Olt  and  a  general  table  prepared.  Table 
XIV  is  table  for  coefficients  for  a  spiral  up  to  15  chord 
lengths  for  use  with  the  instrument  at  any  -chord  point. 

122.  For  the  instrument  at  the  P.S.,  multiply  the  co- 
efficient in  the  columns  headed  0  opposite  the  chord  point 
to  be  located  by  the  value  of  the  spiral  deflection  angle 
for  a  single  chord  length  (unit  spiral  deflection  angle  0J. 


68  UNIFORM   CHORD  LENGTH    METHOD 

123.  To  find  the  deflection  angle  from  the  tangent  at 
any  chord  point,  enter  the  column  whose  heading  gives 
the  number  of  the  chord  point  at  which  the  instrument  is 
placed  and  take  the  coefficient  opposite  the  number  of  the 
chord  point  to  be  located;  then  multiply  the  spiral  deflec- 
tion angle  for  a  single  chord  length  (0J  by  this  coefficient. 
Thus,  as  in  example  cited  above,  for  the  instrument  at 
5,  the  deflection  angle  from  the  tangent  at  this  point  to 
locate  a  point  8  chord  lengths  from  the  P.S.  is  found  to 
be  540,,  and  to  locate  a  point  3  chord  lengths  from  the 
P.S.   is  260,.     This  table  may  easily  be  extended.     The 
variation  in  the  tabular  differences  in  horizontal,  vertical, 
and  diagonal  directions  is  readily  discerned,  and  if  pre- 
ferred the  method  of  differences  may  be  used  for  calcu- 
lating deflection  angles  for  a  particular  case  in  place  of 
a  multiplication  of  these  coefficients. 

124.  To  end  the  spiral  with  a  fractional  chord.   If  the 

number  of  chord  lengths  is  not  integral,  the  first  and 
succeeding  chords  may  be  made  of  uniform  length  until 
the  last,  one  is  reached  and  the  full  deflection  angle  may 
be  turned  off  for  the  P.C.C.  Thus,  for  218.4  ft.  of 
spiral,  ten  20-ft.  chords  may  be  used  and  the  full  deflec- 
tion angle  turned  off  for  the  remaining  18.4  ft. 

125.  To  begin  the  spiral  with  a  fractional  chord  length. 

In  case  it  is  desired  to  begin  the  spiral  with  a  fractional 
chord  length,  the  following  modification  may  be  made. 
Let  m  be  the  ratio  of  this  fractional  chord  length  to  a  full 
chord  length,  and  Om  be  the  spiral  deflection  angle  from 
the  initial  tangent  for  this  fractional  chord  length,  which 
from  the  general  formula  for  9  may  be  seen  to  be  m2  0r 
Let  Qn+m  be  the  spiral  deflection  angle  from  initial  tangent 
to  locate  a  point  (n  +  m)  chord  lengths  away  (n  an  inte- 
ger and  m  fractional),  and  $w+m  the  deflection  angle  from 
tangent  at  instrument  point  (w'  +  ;w)  chord  lengths  from 


DEFLECTION   ANGLES  69 

P.S.  to  locate  a  point  (n  +  m)chord  lengths  from  P.S. 
Substituting  in  formula  (9)  page  8, 

=  On  +  n    (2m   0J  +  Om (9") 

126.  In  the  last  member  of  equation  (9"),  the  first 
term  is  the  spiral  deflection  angle  for  n  full  chords,  the 
second  term  is  n  times  a  constant,  and  the  third  term  is 
the  spiral  deflection  angle  for  the  fractional  chord.  The 
calculations  may  be  simplified  by  the  method  of  differ- 
ences. 

For  example,  for  a  chord  length  of  20  ft.  let  the  be- 
ginning chord  be  8.4  ft.  Then  w  = -^r- =  .42.  If  1.2' be 

zo 

the  unit  spiral  deflection  angle  0lf  0OT=.21'.  To  locate  a 
point  88.4  ft.  from  P.S.,  the  spiral  deflection  angle  at 
1  .S.  will  be 

6n+m  =  16  x  1.2  +  4  x  2  x  .42  x  .21  +  .2  =  23'.4. 
Use  Table  XIV  in  calculating  0n. 


127.  For  the   instrument   at   a  chord  point    (n' 
chord  lengths  from  the  P.S.,  the  deflection  angle  from  the 
tangent  at  this  chord  point  to  locate  a  chord  point   (w+ 
*w)   chord  lengths  from  the  P.S.  is  found  from  equation 
(10)  page  11  to  be 

$n+w=C3(n/  +  w)     (n  —  n')±(n—  n')2]    ^ 

=  $»  +  (»  —  «')    (3w  ^) (10") 

In  the  last  member  of  equation  (10")  the  first  term 
is  the  deflection  angle  for  full  chords,  and  the  second 
term  is  (n  —  n')  times  a  constant.  If  n'  is  greater  than 
n,  the  second  term  is  still  added  numerically  to  the  first. 
Tabular  differences  may  also  be  used. 

128.  To  use  equation    (10")    first  calculate  &n  using 
Table  XIV;  then  add  the  last  term.     Thus,  for  a  chord 
lenecth  of  20  ft.  an,d  a  beginning:  chord  of  8.4  ft.  and  a 


YO  UNIFORM   CHORD  LENGTH    METHOD 

unit  spiral  deflection  angle  of  1.2',  with  the  instrument 
88.4  ft.  from  P.S.,  n'  +  m  =  4.42.  To  locate  a  chord  point 
148.4  ft.  from  P.S.  (»  =  7),  the  deflection  angle  from  the 
tangent  is  found  to  be,  taking  the  coefficient  for  ®n  from 
the  column  headed  4, 

3>n+m  =  45  x  1.2'  +  3  x  3  x  .42  x  .21'  =  54.8'. 
To  locate  a  chord  point  48.8  ft.  from  P.S.,  using  the 
same  instrument  point,  (n  =  %,  n'=4:,  w  — .42),  we  have 
for  the  deflection  angle 

3>n+m  =  20  x  1.2  +  2  x  3  x  .42  x  .21'=  24.5'. 

129.  To  illustrate  the  use  of  these  methods,  take  the 
following  examples.    It  is  desired  to  spiral  a  6°  40'  curve 
with  200  ft.  of  spiral,  using  20-ft.  chords,  c  =  .20.  n  =  10. 
Dn  =  6%.     By  equation   (9')  the  spiral  deflection  angle  0X 

9O2 

for  a  20-ft.  chord  is  J  x^-  x  6°  40'  =  lj'  =  0r    The  multi- 

Li 

plication  of  lj'  by  the  coefficients  in  Table  XIV  for 
instrument  at  0  and  for  instrument  at  10,  gives  the  de- 
sired deflection  angles. 

If  it  is  desired  to  connect  a  tangent  with  a  4°  curve 
so  that  the  offset  o  shall  be  5.0  ft.,  proceed  as  follows. 
By  equation  (14)  the  length  of  spiral  is  415.2  ft.  Using 
25-ft.  chords,  the  deflection  angle  in  minutes  is  by  equa- 

1 0  x  4  x  f i V 
tion  (9')—     *     ^=.602  =  0,.    Table  XIV  will  give  the 

TT.-LO^J 

coefficients  for  multiplication,  and  the  fractional  chord 
may  be  left  for  the  last  measurement. 

130.  To   show  the   use   of   fractional   beginning   and 
ending  chords,  consider  that  a  spiral  138.4  ft.  long  is  to 
connect  with  a  10°   curve  and  that  it  is  desired  to  use 
15-ft.    chords   but   that   the   first   chord    shall    be   9.3    ft. 

0    O 

c=.15.  w--^.==.62.  By  equation  (9'),  the  spiral  deflec- 

-LO 

tion  angle  for  a  15-ft.  chord  0±  is  found  to  be  If',  and  for 


DEFLECTION    ANGLES 


71 


the  point  9.3  ft.  from  P.S.  Om  is  .622  x  1|  =  .62'.  The  table 
below  gives  values  for  field  work,  considering  that  P.S. 
is  at  Sta.  322  +  13.7.  In  the  column  headed  "number  of 


Survey 
Station 

No.  of 
Chord 
Point 

Central 
Angle 

INSTRUMENT  POINT 

P.S. 

0.62 

1.62 

4.62 

5.62 

322+13.7 

P.S. 

0 

0 

1.2' 

1°09.4' 

+23 

0.62 

1.8' 

0.6' 

0 

1°04.1' 

+38 

1.62 

12.8' 

4.3' 

4  6' 

0 

53.0' 

+53 

2.62 

33.5' 

11.2' 

12.5' 

38.6' 

+68 

3.62 

1°03.9' 

21.3' 

23.7' 

20.9' 

+83 

4.62 

1°44.1' 

34.7' 

0 

+98 

5.62 

2°34.0' 

51.3' 

24.1' 

0 

323+13 

6.62 

3°33.6' 

1°11.2' 

+28 

7.62 

1°34.3' 

+43 

8.62 

2°00.7' 

+52.1 

End 

6°55.3' 

2°18.4' 

chord  point"  the  integer  of  the  number  is  n  and  the  frac- 
tional part  is  m.  In  the  succeeding  column  headings  for 
the  instrument  points,  the  integer  is  n'.  Thus,  to  deter- 
mine the  deflection  angle  with  the  instrument  at  0  to 
locate  323  +  43,  by  equation  (9")  and  Table  XIV, 
On+m  =  (64  x  If)  +  (8  x  2.01)  +  .6  =--  2°  00.7'. 

To  determine  the  deflection  with  instrument  at  322  +  83, 
..(«'  =  4),  to  locate  322  +  68,   (n  =  3),  by  equation  (10"), 
3>n+m=  (llxlf)  +  (1x3.02)  =20.9'. 

131.  To  run  in  the  spiral  from  the  P.C.C.,  this  method 
may  likewise  be  used  if  the  P.P.C.  is  at  n  or  at  n  +  m 
chord  lengths  from  the  P.S.  If  it  is  not,  that  is,  if  both 
the  first .  and  last  chord  lengths  are  to  be  fractional,  the 
points  on  the  spiral  as  far  as  the  next  instrument  point 
may  be  set  by  the  principle  that  deflection  angles  will 
equal  the  difference  between  the  deflection  angle  for  a 
circular  curve  and  the  deflection  angle  for  a  spiral  from 


72  STREET  RAILWAY   SPIRALS 

initial  tangent,  both  for  a  distance  equal  to  the  distance 
to  the  desired  point.  After  the  next  instrument  point  is 
reached,  calculate  deflection  angles  as  though  working 
from  the  P.S. 

132.  The  method  of  uniform  chord  lengths  is  subject 
to  the  same  correction  for  0  as  is  given  on  page  9.     As 
it  is  not  likely  that  this  method  will  be  used  for  large 
deflection    angles,   the   error   will   usually   be    negligible. 
For  <£  the  correction  needed  is  almost  exactly  that  for 
the  0  which  enters  into  it;  thus  in  equation  (10')  make 
a  correction  which  would  be  necessary  for  a  0  equal  to 
(n — n')z  0^     This  is  not  often  necessary. 

133.  It  will  be  seen  that  the  method  of  uniform  chord 
lengths  may  have  advantages  where  a  chord  of  full  length 
may  be  used  at  the  beginning  of  the  spiral,  especially 
where  the  rate  a  is  fractional,  and  that  it  is  also  appli- 
cable when  the  beginning  chord  is  fractional.     It  is  more 
especially    applicable    where    evenly    spaced    points    are 
wanted.    Table  XIV  is  a  convenient  table. 


STREET  RAILWAY  SPIRALS 

134.  For  use  in  connection -with  curves  of  short  radii, 
as  street  railway  curves,  the  formulas  for  the  transition 
spiral  may  be  modified  with  advantage.  The  variable 
radius  R  of  the  spiral  may  replace  the  degree-of-curve  D. 
The  product  of  the  radius  at  any  point  by  its  distance 
from  the  P.S.  will  be  shown  to  be  constant  for  a  given 
spiral,  and  this  product  may  be  used  as  the  characteristic 
constant,  taking  the  place  of  a.  The  offset  o  may  be  used 
as  one  fourth  of  the  ordinate  y  of  the  terminal  point  of 
the  spiral  except  for  extreme  lengths.  Certain  other 


THEORY 


approximations  may  be  made  which  are  not  always  allow- 
able with  curves  of  large  radius. 

135,  Theory. — The  general  notation  will  not  be 
changed.  Fig.  16  shows  the  co-ordinates  x  and  y,  spiral 
intersection  angle  A,  spiral  deflection  angle  0,  and  spiral 
tangent-distances  u  and  v  for  a  point  on  the  spiral,  and 


X j 

zj-u^^  -  -^rjjl 1 


PCC, 


FIG,  16. 

also  o  and  t  for  the  full  spiral,  together  with  the  produced 
circular  curve.  As  before,  R  =  radius  of  curvature  at 
any  point  and  s  =  length  of  the  spiral  arc  in  feet  from 
P.S.  to  any  point  of  the  spiral.  From  equation  (1)  page  6, 

D  _  .   100D        573000 


Hence 


L  s 

,    573000 


s  R 


",  a  constant. 


Represent  this  constant  product  of  ^  and  R  by  k.    Then 


d7?=_ 
a  s 

The  property  of  the  transition  spiral  that  R  varies  in- 
versely as  the  distance  along  the  spiral  is  satisfied  by  this 
equation. 


74  STREET    RAILWAY    SPIRALS 

136,  Modification    of    the    formulas    already    derived 
may  be  made  as  follows.     The  angle  subtended  by  a  cir- 
cular arc  in  degrees  is  equal  to = — '— ~m     Since  A 

v     R       R 

is  one  half  as  much  as  the  angle  of  the  same  length  of 
circular  curve  having  a  radius  equal  to  the  terminal 
radius  of  the  spiral, 

28.65  j_  28.65  / 

A  =  ~^~        ~~k~ (* 

This  may  also  be  derived  directly  from  equation  (2)  page 
6  by  substitution  for  the  values  of  a  and  L.    Also 
9.55  s      9.55  r 

*=*A=-/r:  — (53) 

For  large  angles,  if  the  precise  values  of  0  are  desired, 
the  corrections  given  in  the  table  on  page  9  may  be 
made.  However,  for  the  short  distances  involved  this 
correction  may  generally  be  neglected. 

137.  Consider  that  0  =  J  yv  using  the   subscript  1  to 
designate  the  y,  etc,  of  the  terminal  point  of  the  spiral. 
By  trigonometry  y1  —  o  =  R^  vers  A.    Then  yl=^Ri  vers 
A  and 

0  =  1  R±  vers  A ...(54) 

Also  by  substitution  for  a  and  L  in  eauations  (6),  (7), 
(13),  (17),  (21)  and  (23),  pages  17  and  18,  the  follow- 
ing formulas  are  obtained: 


V 

_    s 

(55) 

J 

-v- 

§k 

/                  9   y2 

(56) 

f~l 

40£2C           10  5 

(57) 

24  k 

-1-    Ci 

r  24  & 

•  •  •  "  •  °  •  •  •  •  •  v  u  *  ) 

.  .  (58) 

TABLES  75 


,5 

v=i  j+  —  ±_  ........................  .  .  ____  (60) 

120  k* 

In  the  last  three  equations,  note  that  the  t  correction 
is  J  of  the  x  correction  (last  term  in  equation  (56)  used 
in  finding  x  from  s)  ;  the  C  correction  is  ^  of  the  x 
correction;  and  the  v  correction  is  J  of  the  x  correction. 

For  extreme  cases,  the  values  of  y  and  o  given  by 
equations  (55)  and  (57)  will  be  slightly  too  large.  For 

y,   subtract   .003  —  g-  from   the   results   of  equation    (55). 
k 

For  o  subtract  one  eighth  as  much.  For  x,  t,  C,  and  v, 
the  terms  given  in  the  equations  above  will  generally  be 
sufficiently  accurate. 

138.    The  following  equations  may  be  repeated  here: 
T  =  t+(R1  +  o)   fan  J  /  ......................  (18) 

E=(Ri  +  o)   exsec  1  I  +  o  .........  ......  .....  (20) 

u  =  x  —  v  cot  A  .... 


.  (22) 

=  x  —  v  cos  A. 

139.  The  Tables — Tables' \v ill  .facilitate  the  applica- 
tion of  these  equations.  Tables  XV-XIX  give  properties 
of  five  spirals.  The  spiral  is  to  be  used  up  to  that  length 
which  gives  the  required  radius.  The  x  correction  is  the 
amount  to  be  subtracted  from  the  length  of  the  spiral  to 
give  the  abscissa  x.  The  long  chord  C  may  be  found  by 
subtracting  four  ninths  of  the  x  correction  from  the 
length  of  spiral.  Similarly  the  spiral  tangent-distance  v 
is  found  by  adding  one  third  of  the  x  correction  to  one 
third  of  the  length  of  spiral.  Interpolations  for  distances 
between  those  given  in  the  tables  may  be  made,  but  it  is 
best  to  compute  R  and  the  angles. 


76  STREET    RAILWAY    SPIRALS 

140.  Laying  out  spiral.— The    same    methods    may    be 
used  in  laying  out  spirals   of  short  radii   as   have  been 
described  for  curves  of  large  radii.     The  location  of  the 
P.S.,  P.C.C.,  and  P.C.   is  generally  not  difficult.     If  the 
lines  have  been  run  to  an  intersection,   as   is   generally 
desirable,   the   tangent-distance    T   may   be   measured   to 
locate  the  P.S.     The  P.C.C.  may  be  located  by  turning 
off   the    full   spiral   deflection   angle   0,,    at  the   P.S.   and 
measuring  the  long  chord  C;  or  the  spiral  tangent  dis- 
tances u  and  v  may  be  calculated  and  the  angle  A±  turned 
at  their  intersection  point.     In  either  case  the  tangent  at 
the  P.C.C.  may  readily  be  found.     Another  method  is  to 
locate  the  P.C.  by  offsetting  the  distance  o  from  the  ini- 
tial tangent  and  then  running  in  the  circular  curve  to  the 
P.C.C. 

141.  Centers  for  the  spiral  may  be  set  (a)  by  meas- 
uring   ordinates    from    the    initial    tangent    for    the    full 
length  of  spiral;  (b)  by  ordinates  from  the  initial  tangent 
as  far  as  the  middle  of  the  spiral  and  from  the  produced 
circular  curve  for  the  remaining  half  length;  (c)  by  ordi- 
nates  from  the   initial  tangent  for  about  two  thirds   of 
the  spiral  and  from  the  terminal  spiral  tangent  for  the 
remainder  of  the   spiral.     The  offsets   from  the  circular 
curve   will   be   the    same    for   given   distances    from   the 
P.C.C.  as  the  y  for  an  equal  length  of  spiral.    The  offsets 
from  the  terminal  spiral  tangent  will  be  the   difference 
between  the  offset  for  the  circular  curve  and  the  y  for 
a  spiral,  both   for   a  length   equal  to  the  distance   from 

2  2 

P.C.C.  to  the  point  located.    This  will  be —y  where 

s  is  the  distance  of  the  point  from  the  P.C.C.  and  R  is 
the  radius  of  the  circular  curve. 

142.  Location    by   means    of    deflection    angles    from 
initial  tangent  at  P.S.  is  so  similar  to  that  for  railway 


ARC  EXCESS  77 

spirals  already  described  that  it  need  not  be  further  dis- 
cussed. If  the  rails  have  previously  been  bent  to  their 
proper  curvature  very  few  centers  need  be  set. 

143.  Arc  excess. — It  must  be  borne  in  mind  that 
the  actual  length  of  arc  is  considered  in  the  formulas 
here  given,  and  care  must  be  taken  to  provide  for  the 
difference  between  arc  and  chord  measurement.  The 
long  chord  is  easily  found  by  the  x  correction  of  the 
tables  as  already  indicated.  For  other  chords  of  spiral 
arcs  (not  from  P.S.),  it  will  be  sufficiently  accurate  to 
use  for  the  excess  of  arc  over  chord  the  excess  of  the 
same  length  of  circular  arc  having  a  radius  equal  to  that 
of  the  middle  point  of  the  spiral  arc  under  consideration. 
The  excess  length  of  a  circular  arc  over  its  chord  may 


be   calculated  from   the   approximate   formula     — — — — 

where  c  may  be  used  as  the  length  of  either  chord  or  arc. 
It  may  also  be  noted  here  that  the  number  of  degrees 

of  angle  in  a  circular  arc  is   — =~     and  the  deflection 

J\. 

angle  from  tangent  is  of  course  half  of  this. - 

144.  Curving  rails. — The  principles  here  outlined  are 
for  the  center  line  of  track,  but  it  may  be  desirable  to 
have  measurements  for  the  curves  formed  by  the  rails 
for  use  in  curving  rails,  etc.  Although  the  outer  and 
inner  rails  will  be  parallel  to  the  center  line,  their  lines 
will  not  be  true  spirals,  and  allowance  should  be  made 
for  this.  For  data  in  bending  rails,  it  will  be  well  first 
to  get  the  variation  in  length  of  rail  from  the  length  of 
the  center  line.  For  a  given  point  on  the  center  line, 

first  find  the  A  of  the  spiral.    The  outer  rail  will  be    : 

180 

TT  x  %G   longer   than   the   center   line,    and   the   inner   rail 
will  be   as  much   shorter,   G  being  the   gauge   of  track. 


78  STREET    RAILWAY    SPIRALS 

Thus  for  £  =  1500  and%a  spiral  distance  of  30  ft.  A  =  17° 

17  18 
11'  and  the  excess  length  in  outer  rail  is     — - —     x  3.14  x 

180 

2.35  =  .70  ft.  The  outer  rail  distance  will  be  30.70  ft. 
and  the  inner  rail  distance  29.30  ft.  The  ordinate  of  the 
rail  from  its  own  initial  tangent  will  be  the  3;  for  the 
center  line  spiral  d=  J  G  vers  A ;  plus  to  be  used  for  the 
outer  rail  and  minus  for  the  inner  one.  Thus  for  the 
example  above  cited,  the  ordinates  for  the  rails  opposite 
a  center  distant  30  ft.  from  the  P.S.  (30.70  along  outer 
rail  and  29.30  ft.  along  the  inner  rail)  will  be  3.00  ±  .11 
=  3.11  and  2.89.  In  locating  points  on  the  rails,  allow- 
ance should  be  made  for  the  difference  between  the  cen- 
ter line  distance  and  the  rail  distance.  The  x  for  the 
Doint  on  the  rail  will  be  the  x  for  the  corresponding  point 
on  the  center  spiral  d=  \  G  sin  A.  These  principles  will 
apply  to  any  point  on  the  spiral.  It  will  be  well  to  tabu- 
late values  for  the  sets  of  curves  most  used. 

145.  If  it  is  desired  to  locate  the  last  third  of  the 
spiral  from  the  terminal  spiral  tangent  for  the  two  rails, 
the  length  and  position  of  these  tangents  may  be  calculated 
from  tneir  ordinates  and  the  A  already  used,  and  points 
on  the  rails  located  by  offsets  from  these  tangents.    These 
offsets  may  readily  be  calculated  by  the  principles  already 
outlined. 

146.  Double  track. — Double  track  curves  will  gen- 
erally need  radii  of  different  lengths.   The  spirals  for  both 
curves  may  be  taken   from  the   same   table,   or  the   one 
for  the  outside  curve  may  be  taken  from  the  table  hav- 
ing a  value  of  k  next  higher  than  that  used  with   the 
inside  curve.     In  the  latter  case  the  two  spirals  will  be 
of  nearly  the  same  length  and  their  ends  will  be  nearly 
opposite.      If  the .  distance   between   center   lines   on   the 
curve  be  made  equal  to  the  distance  between  center  lines 


DOUBLE    TRACK  79 

on  tangent  plus  the  difference  in  the  o  for  the  inside  and 
outside  spiral,  the  circular  parts  of  the  two  curves  will 
be  parallel  and  have  the  same  center,  and  the  radius  of 
the  outside  curve  will  be  equal  to  the  radius  of  the  inside 
curve  plus  the  distance  between  tracks  on  the  curve. 

147.  In  case  consideration  of  clearance  requires 
greater  distance  between  the  tracks  on  curves  than  on 
tangents,  care  must  be  exercised  in  the  selection  of  the 
spiral.  The  calculation  of  the  external-distance  E  will 
best  enable  the  distance  between  center  lines  at  their 
middle  points  to  be  determined.  If  the  inside  radius  is 
assumed,  first  find  its  external  distance  E,  to  this  add  the 
distance  from  one  P.  I.  to  the  other,  and  subtract  the 
required  distance  between  the  curves.  The  remainder 
will  be  the  external-distance  E  for  the  .outside  curve  from 
which  the  desired  radius  may  be  found.  As  by  this  ar- 
rangement the  two  curves  will  be  closer  at  their  ends 
than  at  their  middle,  care  must  be  taken  to  secure  suffi- 
cient clearance.  The  selection  of  curves  and  spirals  for 
double  track  is  more  complex  than  for  single  track. 


CONCLUSION 

148.  Besides  the  problems  and  methods  here  pre- 
sented, many  other  applications  may  be  made.  For  par- 
ticular conditions  the  engineer  may  develop  speci-l 
methods. 

The  preceding  methods  generally  have  been  based  upon 
the  principle  that  the  spiral  is  to  have  the  same  degree- 
of-curve  at  the  end  as  the  main  curve,  and  slight  modi- 
fications may  be  necessary  when  not  so.  The  value  of  o 
and  of  the  angle  in  the  circular  curve  omitted  must  be 
that  for  the  spiral  used.  Thus  with  a  =  2  the  spiral  at  the 


80  CONCLUSION 

end  of  300  ft.  will  be  a  6°  curve.  It  may,  however,  be 
there  compounded  with  a  curve  of  different  radius,  as  a 
6°  30'  curve,  provided  the  offset  is  3.93  and  the  central 
angle  between  the  P.C.  and  the  P.C.C.  is  9°.  Generally, 
in  order  to  utilize  the  problems  given  for  old  track,  etc., 
the  formulas  will  need  to  be  modified  if  D0  does  not 
agree  with  Dr 

149.  In  field  work  most  of  the  usual  formulas  of  the 
various  location  problems,  like  ''Required  to  change  the 
P.C.  so  that  the  curve  may  end  in  a  parallel  tangent," 
may  be   used   without   modification   with   curves   having 
transition  endings,  by  simply  considering  the  whole  inter- 
section  angle   including  the  angle   in  the   spirals.     This 
is  true  whenever  the  same  amount  of  spiral  is  used  with 
the  new  curve. 

If  the  degree-of-curve  changes  and  with  it  the  length 
of  the  spiral,  the  difference  between  the  o's  in  the  two 
cases  must  be  allowed  for.  With  a  little  practice  in  using 
such  formulas  with  spirals,  the  engineer  will  find  no 
difficulty. 

150.  The  transition  spiral  has  the  merit  of  compara- 
tive  simplicity  and  extreme  flexibility.     It  is  a  natural 
method,  since  it  is  so  similar  to  the  methods  used  in  lay- 
ing out  circular  curves.     Like  circular  curves,  the  length 
along  the  curve  is  the  principal  term,  and  the  degree-of- 
curve,  central  angle,  deflection  angles  and  ordinates  are 
obtainable  from  this  variable.     It  may  be  used  with  any 
main   curve,   even   of    fractional   degree;    any   length   of 
chord  may  be  used  in  measurement  under  the  same  re- 
strictions as  circular  curves,  and  as  it  is  not  necessary  to 
restrict   the   measurements   to    a   common   chord   length, 
intermediate  points  may  be  readily  located.     The  calcula- 
tions for  angles  and  distances  are  easily  made.     If  de- 
sired, the  tangent  and  the  circular  curve  may  be  run  out 


CONCLUSION  81 

and  the  spiral  put  in  by  co-ordinates,  one  half  from  the 
tangent  and  one  half  from  the  circular  curve.  This  is 
especially  applicable  to  location  work  and  to  short  spirals. 
The  engineer  should  not  be  frightened  by  the  mathe- 
matics in  the  demonstration  of  the  formulas;  the  prin- 
ciples and  methods  may  be  understood  without  mastering 
the  demonstrations.  Experience  has  shown  that  the  ordi- 
nary transit  man,  with  a  little  thought  and  study,  can 
understand  and  use  the  transition  spiral  as  easily  as  cir- 
cular curves,  and  that  young  assistants  without  previous 
training  readily  take  up  the  work. 

151.  With  reference  to  the  use  of  the  cubic  parabola 
as  an  easement  it  may  be  said  that,  except  for  the  rela- 
tion between  x  and  y,  it  has  no  properties  of  value  for 
a  transition  curve  which  are  not  merely  approximations 
of  the  transition  spiral.     Within  small  limits,  the  radius 
of  curvature  and  the  angles  to  be  used  approach  some- 
what closely  to  those  for  the  transition  spiral.     As  soon 
as  x  differs  materially  from  the  length  of  curve,  a  cor- 
rection has   to  be  made.     The   radius  of  the   curvature 
finally  begins  to  increase.    The  investigation  of  the  cubic 
parabola  in  reference  to  its  radius  of  curvature,  its  angle 
turned,   the   angular  deflection  to  points  on  it,   and  the 
length  of  the  curve,  require  as  long  mathematical  equa- 
tions as  those  governing  the  transition  spiral.     Many  at- 
tempts have  been  made  to  utilize  this  curve,  but  both  field 
work  and  computations  are  too  intricate  and  inconvenient 
if  the  curve  has  any  considerable  length,  and  it  has  no 
advantage  over  the  transition  spiral. 

152.  The  question  of  the  efficiency  of  easement  curves 
is  of  considerable  importance.     The  objection  is   some- 
times raised  that  even  if  track  is  laid  out  with  a  carefully 
fitted  spiral  there  would  be  no  possibility  of  keeping  it  in 
place  by  the  methods  of  the  ordinary  trackman.     This 


82  CONCLUSION 

identical  objection  could  be  made  with  the  same  force 
against  carefully  laid  out  circular  curves,  yet  no  engineer 
would  recommend  abolishing  that  practice.  Even  if,  in 
re-lining,  the  transition  curve  is  considerably  distorted, 
it  remains  an  easement,  and  will  be  in  far  better  riding 
condition  than  a  distorted  circular  curve.  By  marking 
the  P.S.  and  the  P.C.C  with  a  stake  or  post,  with  inter- 
mediate points  on  long  spirals,  the  trackman  will  be  able 
to  keep  the  spiral  in  as  good  condition  as  though  it  were 
of  uniform  curvature.  The  short  spirals  advocated  by 
some  engineers  have  proved  to  be  insufficient.  For  effi- 
cient service,  a  length  of  spiral  which  will  give  an  o  of 
considerable  amount  must  be  used,  even  if  this  necessi- 
tates widening  the  roadbed. 

153.  Properly  constructed  spirals  would  frequently 
allow  the  use  of  sharper  curvature — since  the  riding  qual- 
ity of  curves  may  be  the  governing  consideration  in  the 
selection  of  a  maximum — and  thus  make  a  saving  in 
construction.  By  fitting  curves  with  proper  transition 
spirals,  roads  using  sharp  curves  may  partially  relieve  the 
objection  of  the  public  to  traveling  by  their  routes.  The  in- 
troduction of  fast  trains  has  made  it  necessary  to  take  every 
precaution  to  secure  an  easy-riding  track.  The  disagree- 
able lurch  and  necessary  "slow  order"  for  fast  trains  at 
certain  curves  on  many  roads  has  been  entirely  eliminated 
by  the  construction  of  proper  spirals,  and  passengers  do 
not  now  know  when  such  curves  are  reached.  The 
transition  curve  has,  then,  a  financial  value  largely  over- 
balancing its  cost.  The  adoption  of  such  curves  by  many 
of  our  principal  railways  proves  their  efficiency,  and  the 
future  will  see  a  much  more  general  adoption. 


EXPLANATION.    OF    TABLES 


EXPLANATION  OF  TABLES. 


In  Tables  I-XI,  the  columns  give  the  following  prop- 
erties : 

1.  The  distance  in  feet  from  the  P.S.  along  the  spiral 
to  a  point  on  the  spiral;  i.  e.,  100  L.    The  full  length  of 
spiral  will  give  values  for  the  terminal  point,  the  P.C.C. 
of  main  curve. 

2.  D,  the  degree-of-curve  of  the  spiral  at  any  point. 
It  becomes  D±  at  the  P.C.C. 


^  -R\C. 

* 


^  >\PC.C. 


FIG.  17. 

3.  A,  the  spiral  angle  or  change  of  direction  of  the 
spiral  to  the  point. 

4.  0,  the  spiral  deflection  angle  at  the  P.S.  from  the 
initial  tangent  to  locate  the  point. 

5.  o,  the  offset  from  the  initial  tangent  to  the  P.C. 
of  main  curve  produced  backward.     Enter  the  table  with 
the  full  length  of  the  spiral  used. 

6.  y,  the  ordinate  from  the  initial  tangent  as  the  axis 
of  X." 


EXPLANATION   OF  TABLES  85 

7.  x  COR.,  an  amount  to  be  subtracted  from  the  dis- 
tance in  feet  from  the  P.S.  along  the  spiral  to  find  the 
abscissa,  x,  of  the  point,    x  =  100  L  —  x  COR. 

8.  t  COR.,  an  amount  to  be  subtracted  from  half  the 
full  length  of  the  spiral  in  feet  to  find  t,  the  abscissa  of 
the  P.C.     Enter  the  table  with  full  length  of  the  spiral 
used.    t=^L  —  t  COR. 

To  find  the  long  chord  to  P.S.,  subtract  four  ninths 
(.444)  of  x  COR.  from  the  length  of  the  spiral  in  feet. 
C  =  100  L  —  f  x  COR.  For  chords  not  ending  at  P.S.,  see 
pages  16  and  77. 

To  find  the  terminal  spiral  tangent-distance,  add  one 
third  x  COR.  to  one  third  the  spiral  distance  to  the  point. 
z,=  i|o.L  +  J  x  COR. 

Intermediate  values  may  be  found  by  interpolation. 

With  transit  at  intermediate  point  on  spiral,  for  deflec- 
tion angle  <£,  see  pages  10,  20,  and  40. 

To  use  Table  IV  for  other  values  of  a,  multiply  the 
tabulated  values  of  D9  A,  9,  o,  and  y  in  Table  IV  opposite 
the  given  distance  from  the  P.S.  by  the  a  of  the  desired 
spiral,  and  x  COR.  and  t  COR.  by  the  square  of  a.  For 
inaccuracies  of  this  method  see  page  27.  If  a  and  D  or  o 
are  given,  first  find  L. 

Table  XII  permits  ordinates  to  be  calculated  from  o. 
See  Fig.  5. 

For  the  use  of  Tables  XIII  and  XIV  see  page  67,  and 
for  Tables  XV,  XIX,  see  page  75. 

Table  XX  gives  values  of  o  and  L  for  values  of  a 
and  D. 

For  full  nomenclature,  see  page  3. 

For  equations  and  summary  of  principles,  see  page  17. 

For  fuller  explanation  of  tables  and  errors  of  interpo- 
lation, see  page  24. 

For  choice  of  a,  see  page  28. 


1°  in  200  ft. 


TABLE  I.     TRANSITION  SPIRAL. 


Length 

D 

A 

9 

0 

y 

x  COR. 

/COR. 

25 

Q°Q7}4' 

o°oo:9 

o°oo:3 

.00 

.00 

50 

0  15 

0  03.8 

0  01.3 

.00 

.02 

75 

0  22^ 

0  08.4 

0  02.8 

.02 

.06 

100 

0  30 

0  15. 

0  05. 

.04 

.15 

125 

0  37X 

0  23.4 

0  07  8 

.07 

.29 

150 

0  45 

0  33.8 

0  11.3 

.12 

.49 

175 

0  52/2 

0  45.9 

0  15.3 

.20 

.78 

.00 

200 

1  00 

1  00. 

0  20. 

.29 

1.16 

.01 

225 

1  07^ 

1  15.9 

0  25.3 

.41 

1.66 

.01 

250 

1  15 

1  33.8 

0  31.3 

.57 

2.27 

.02 

.00 

275 

1  22X 

1  53.4 

0  37.8 

.76 

3.03 

.03 

.01 

300 

1  30 

2  15. 

0  45. 

.98 

3.93 

.05 

.01 

325 

1  37^ 

2  38.4 

0  52.8 

1.25 

5.00 

.07 

.01 

350 

1  45 

3  03.8 

1  01.3 

1.56 

6.23 

.10 

.02 

375 

1  52^ 

3  30.9 

1  10.3 

1.92 

7.67 

.14 

.02 

400 

2  00 

4  00. 

1  20. 

2.33 

9.31 

.19 

.03 

425 

2  07^ 

4  30.9 

1  30.3 

2.79 

11.16 

.26 

.04 

450 

2  15 

5  03.8 

1  41.3 

3.31 

13.25 

.35 

.06 

475 

2  22/2 

5  38.4 

1  52.8 

3.89 

15.58 

.46 

.08 

500 

2  30 

6  15. 

2  05. 

4.54 

18.16 

.59 

.10 

525 

2  37^ 

6  53.4 

2  17.8 

5.26 

21.03 

.75 

.13 

550 

2  45 

7  33.8 

2  31.3 

6.04 

24.17 

.95 

.16 

575 

2  52/2 

8  15.9 

2  45.3 

6.91 

27.62 

1.20 

.20 

600 

3  00 

9  00. 

3  00. 

7.84 

31.36 

1.48 

.24 

'625 

3  07^ 

9  45.9 

3  15.3 

8.87 

35.45 

1.81 

.30 

650 

3  15 

10  33.8 

3  31.3 

9.97 

39.85 

2.21 

.37 

675 

3  22X 

11  23.4 

3  47.8 

11.16 

44.63 

2.66 

.44 

700 

3  30 

12  15. 

4  04.9 

12.45 

49.73 

3.20 

.53 

725 

3  37^ 

13  08.4 

4  22.7 

13.83 

55.22 

3.81 

.64 

750 

3  45 

14  03.8 

4  41.2 

15.30 

61.09 

4.51 

.75 

775 

3  52K 

15  00.9 

5  00.1 

16.88 

67.37 

5.31 

.89 

800 

4  00 

16  00. 

5  19.8 

18.56 

74.05 

6  22 

1.04 

1°  in  150  ft. 


TABLK  II.     TRANSITION  SPIRAL. 


Length 

D 

A 

9 

0 

y 

x  COR. 

/COR. 

25 

0°10' 

0°01'.3 

0°00'4 

.00 

.00 

50 

0  20 

0  05. 

0  01.7 

.01 

.02 

75 

0  30 

0  11.3 

0  03.8 

.02 

.08 

100 

0  40 

0  20. 

0  06.7 

.05 

.19 

125 

0  50 

0  31.3 

0  10.4 

.10 

.38 

150 

1  00 

0  45. 

0  15. 

.16 

.65 

0.00 

175 

1  10  ^ 

1  01.3 

0  20.4 

.26 

1.04 

0.01 

200 

1  20 

1  20 

0  26.7 

.39 

1  55 

0  01 

TABLE  II.— Continued. 


1°  in  150  ft. 


Length 

D 

A 

8 

0 

y 

x  COR. 

/COR. 

225 

1°30' 

i°4i:3 

0°33'.8 

.55 

2.21 

.02 

.00 

250 

1  40 

2  05. 

0  41.7 

.76 

3.03 

.03 

.01 

275 

1  50 

2  31.3 

0  50.4 

1.01 

4.04 

.05 

.01 

300 

2  00 

3  00. 

1  00. 

1.31 

5.23 

.08 

.01 

325 

2  10 

3  31.3 

1  10.4 

1.66 

6.66 

.12 

.02 

350 

2  20 

4  05. 

1  21.7 

2.08 

8.31 

.18 

.03 

375 

2  30 

4  41.3 

1  33.8 

2.56 

10.23 

.25 

.04 

400 

2  40 

5  20. 

1  46.7 

3.10 

12.40 

.35 

.06 

425 

'2  50 

6  01.3 

2  00.4 

3.72 

14.88 

.47 

.08 

450 

3  00 

6  45. 

2  15. 

4.41 

17.66 

.62 

.10 

475 

3  10 

7  31.3 

2  30.4 

5.19 

20.76 

.82 

.14 

500 

3  20 

8  20. 

2  46.7 

6.05 

24.20 

1.06 

.18 

525 

3  30 

9  11.3 

3  03.8 

7.01 

28.02 

1.35 

.22 

550 

3  40 

10  05. 

3  21.7 

8.05 

32.19 

1.70 

.28 

575 

3  50 

11  01.3 

3  40.4 

9.20 

36.78 

2.12 

.36 

600 

4  00 

12  00. 

3  59.9 

10.45 

41  76 

2.63 

.44 

TABLK  III.     TRANSITION  SPIRAL. 


1°  in  125  ft. 


Length 

D 

4 

0 

0 

y 

x  COR. 

t  COR. 

25 

0°12' 

0°01^' 

0°00^' 

.00 

.00 

50 

0  24 

0  06 

0  02 

.01 

.03 

75 

0  36 

0  13^ 

0  04)4 

.02 

.10 

100 

0  48 

0  24 

0  08 

.06 

.23 

125 

1  00 

0  37^ 

0  12^ 

.11 

.46 

150 

1  12 

0  54 

0  18 

.20 

.79 

.00 

175 

1  24 

1  13^ 

0  24^ 

.31 

1.25 

.01 

200 

1  36 

1  36 

0  32 

.47 

1.86 

.02 

225 

1  48 

2  01^ 

0  40^ 

.66 

2.65 

.03 

.00 

250 

2  00 

2  30 

0  50 

.91 

3.64 

.05 

.01 

275 

2  12 

3  01/2 

1  00^ 

1.21 

4.84 

.08 

.01 

300 

2  24 

3  36 

1  12 

1.57 

6.28 

.12 

.02 

325 

2  36 

4  13^ 

1  24^ 

2.00 

7.99 

.18 

.03 

350 

2  48 

4  54 

1  38 

2.49 

9.97 

.26 

.04 

375 

3  00 

5  37^ 

1  52^ 

3.07 

12.27 

.36 

.06 

400 

3  12 

6  24 

2  08 

3.72 

14.88 

.50 

.08 

425 

3  24 

7  13K 

2  24^ 

4.47 

17.85 

.68 

.11 

450 

3  36 

8  06 

.2  42 

5.31 

21.18 

.90 

.15 

475 

3  48 

9  01X 

3  00^ 

6.23 

24.90 

1.18 

.20 

500 

4  00 

10  00 

3  20 

7.26 

29.02 

1.52 

.25 

TABLE  IV.     TRANSITION  SPIRAL 
in  100  ft. 


Length 

D 

A 

e 

0 

y 

x  COR. 

/COR. 

10 

0.1° 

o°oo:3 

o°oo:i 

.000 

.000 

.000 

.000 

20 

0.2 

01.2 

00.4 

.001 

.002 

30 

0.3 

02.7 

00.9 

.002 

.008 

40 

0.4 

04.8 

01.6 

.005 

.019 

50 

0.5 

07.5 

02.5 

.009 

.036 

60 

0.6 

0  10.8 

0  03.6 

.016 

.063 

70 

0.7 

14.7 

04.9 

.025 

.100 

80 

0.8 

19.2 

06.4 

.037 

.149 

90 

0.9 

24.3 

08.1 

.053 

.212 

100 

1.0 

30. 

10. 

.073 

.291 

.001 

110 

1.1 

0  36.3 

0  12.1 

.097 

.387 

.001 

120 

1.2 

43.2 

14.4 

.126 

.503 

.002 

130 

1.3 

50.7 

16.9 

.160 

.639 

.003 

140 

1.4 

58.8 

19.6 

.199 

.798 

.004 

150 

1.5 

1  07.5 

22.5 

.245 

.982 

.006 

.001 

160 

1.6 

1  16.8 

0  25.6 

.298 

1.191 

.008 

.001 

170 

1.7 

1  26.7 

28.9 

.357 

1.429 

.011 

.002 

180 

1.8 

1  37.2 

32.4 

.424 

1.696 

.014 

.002 

190 

1.9 

1  48.3 

36.1 

.499 

1.995 

.019 

.003 

200 

2.0 

2  00. 

40. 

.582 

2.327 

.024 

.004 

210 

2.1 

2  12.3 

0  44.1 

.673 

2.690 

,031 

.005 

220 

2  2 

2  25.2 

48  4 

.774 

3.097 

.039 

.006 

230 

2.3 

2  38.7 

52.9 

.885 

3.538 

.049 

.008 

240 

2  4 

2  52.8 

57.6 

1.005 

4.020 

.061 

.010 

250 

2.5 

3  07.5 

1  02.5 

1.136 

4.544 

.074 

.012 

260 

2.6 

3  22.8 

1  07.6 

1.278 

5.111 

.090 

.015 

270 

2.7 

3  38.7 

1  12.9 

1.431 

5.724 

.109 

.018 

280 

2.8 

3  55.2 

1  18.4 

1.596 

6.383 

.131 

.022 

290 

2.9 

4  12.3 

1  24.1 

1.773 

7.091 

.156 

.027 

300 

3.0 

4  30. 

1  30. 

1.963 

7.850 

.185 

.031 

310 

3.1 

4  48.3 

1  36.1 

2.166 

8.66 

.218 

.036 

320 

3.2 

5  07.2 

1  42.4 

2.382 

9.53 

.255 

.043 

330 

3.3 

5  26.7 

1  48.9 

2.612 

10.45 

.298 

.050 

340 

3.4 

5  46.8 

1  55.6 

2.857 

11.42 

.346 

.058 

350 

3.5 

6  07.5 

2  02.5 

3.116 

12.46 

.400 

.067 

360 

3.6 

6  28.8 

2  09.6 

3.391 

13.56 

.460 

.077 

370 

3.7 

6  50.7 

2  16.9 

3.681 

14.72 

.528 

.088 

380 

3.8 

7  13.2 

2  24.4 

3.988 

15.94 

.603 

.100 

390 

3.9 

7  36.3 

2  32.1 

4.311 

17.23. 

.686 

.114 

400 

4.0 

8  00. 

2  40. 

4.651 

18.59 

.779 

.130 

410 

4.1 

8  24.3 

2  48.1 

5.01 

20.02 

.SSI 

.147 

420 

4.2 

8  49.2 

2  56.4 

5.38 

21.51 

.994 

.166 

430 

4  3 

9  14.7 

3  04.9 

5.78 

23.08 

1.118 

.186 

440 

4.4 

9  40.8 

3  13.6 

6.19 

24.73 

1.254 

*'.2Q? 

450 

4.5 

10  07.5 

3  22.5 

6.62 

26.45 

1.403 

.234 

1°  in  100  ft. 


TABLE  TV.— Continued. 


f 

Length 

D 

J 

(-) 

0 

y 

^•COR. 

/CoR. 

460 

4.6° 

10°34'.8 

3°31'6 

7  07 

28  24 

1.57 

.26 

470 

4.7 

11  02.7 

3  40.9 

7.54 

30.12 

-1.74 

.29 

480 

4.8 

11  31.2 

3  50.4 

8  03 

32.07 

1.94 

.32 

490 

4.9 

12  00.3 

4  00.1 

8.54 

34.11 

2.15 

.36 

500 

5.0 

12  30. 

4  10. 

9  07 

36.23 

2.37 

.40 

516 

5.1 

13  00.3 

4  20.1 

9.63 

38.44 

2.62 

.44 

520 

5.2 

13  31.2 

4  30.4 

10.20 

40.73 

2.89 

.48 

530 

5  3 

14  02.7 

4  40.9 

10.80 

43.12 

3.17 

.53 

540 

5.4 

14  34.8 

4  51.4 

11.42 

45.59 

3.49 

.58 

550 

5.5 

15  07.5 

5  02.3 

12.07 

48.15 

3.82 

.64 

560 

5.6 

15  40.8 

5  13.4 

12.74 

50.83 

4.18 

.70 

570 

5.7 

16  14.7 

5  24.7 

13.43 

53.56 

4.56 

.76 

580 

5.8 

16  49.2 

5  36.2 

14.14 

56.40 

4.98 

.83 

590 

5.9 

17  24.3 

5  47.8 

14.89 

59.34 

5.42 

.90 

600 

6.0 

18  00. 

5  59.7 

15.65 

62.39 

5.89 

.98 

610 

6.1 

18  36.3 

6  11.8 

16.44 

65  52 

6.40 

1.07 

620 

6.2 

19  13.2 

6  24  1 

17.26 

68.77 

6.94 

1.16 

630 

6.3 

19  50.7 

6  36  5 

18.10 

72.11 

-  7.51 

1  25 

640 

64 

20  28.8 

6  49.1 

18.97 

75.56 

8.13 

1.36 

650 

6.5 

21  07.5 

7  02.0 

19.87 

79.11 

8.78 

1.47 

660 

6  6 

21  47. 

7  15.1 

20.79 

82.79 

9.48 

1.57 

670 

6.7 

22  27. 

7  28.5 

21.74 

86.56 

10.22 

1.69 

680 

6.8 

23  07. 

7  41.8 

22.73 

90.43 

11.00 

1.82 

690 

6.9 

23  48. 

7  55.3 

23.73 

94.42 

11.82 

1.96 

700 

7.0 

24  30. 

8  09.3 

24.79 

98.50 

12  70 

2.10 

TABLE  V.     TRANSITION  SPIRAL. 


1°  in  80  ft. 


Length 

D 

J 

9 

o 

y 

x  COR. 

/COR. 

~~T(T 

0°07!' 

0°00!' 

0°00' 

.00 

.00 

.00 

.00 

20 

0  15 

0  01! 

0  00! 

.00 

.00 

30 

0  22! 

0  03! 

0  01 

.00 

.01 

40 

0  30 

0  06 

0  02 

.00 

.02 

50 

0  371 

0  09! 

0  03 

.01 

.04 

60 

0  45 

0  13! 

0  04J 

.02 

.08 

70 

0  52! 

0  18! 

0  06 

.03 

.12 

80 

1  00 

0  24 

0  08 

.05 

.19 

90 

1  07! 

0  30! 

0  10 

.07 

.26 

100 

1  15 

0  37! 

0  12! 

.09 

.36 

110 

1  22J 

0  45! 

0  15 

.12 

.48 

120 

1  30 

0  54 

0  18 

.16 

.63 

130 

1  37! 

1  03! 

0  21 

.20 

.80 

140 

1  45 

1  13! 

0  24! 

.25 

1.00 

150 

1  52! 

1  24! 

0  28 

.31 

1.23 

.00 

.00 

1°  in  80  ft. 


TABLE   V. — Continued. 


Length 

D 

A 

B 

0 

y 

x  COR. 

i 
/COR 

160 

2°00' 

1°36' 

0°32' 

.37 

1.49 

.0 

.0 

170 

2  07* 

1  48* 

0  36 

.45 

1.77 

180 

2  15 

2  01* 

0  40* 

.53 

2.12 

190 

2  22* 

2  15* 

0  45 

.62 

2.50 

200 

2  30 

2  30 

0  50 

.73 

2.90 

210 

2  37* 

2  45* 

0  55 

.84 

3.36 

220 

2  45 

3  01* 

1  00* 

.97 

3.87 

230 

2  521 

3  18* 

1  06 

1.10 

4.42 

240 

3  00 

3  36 

1  12 

1.25 

5.02 

250 

3  07* 

3  54* 

1  18 

1.42 

5.67 

.1 

260 

3  15 

4  13* 

1  24* 

1.59 

6.38 

.1 

270 

3  221 

4  33* 

1  31 

1.79 

7.15 

.2 

280 

3  30 

4  54 

1  38 

1.99 

7.98 

.2 

290 

3  371 

5  15* 

1  45 

2.21 

8.86 

.2 

300 

3  45 

5  37* 

1  52* 

2.45 

9.81 

.3 

310 

3  52* 

6  00* 

2  00 

2.70 

10.74 

.3 

320 

4  00 

6  24 

2  08 

2.98 

11.91 

.4 

330 

4  07* 

6  48* 

2  16 

3.26 

13.06 

.4 

340 

4  15 

7  13i 

2  24* 

3.57 

14.28 

.5 

350 

4  22* 

7  39* 

2  33 

3.89 

15.57 

.6 

360 

4  30 

8  06 

2  42 

4.23 

16.95 

.7 

.1 

370 

4  37* 

8  33* 

2  51 

4.59 

18.40 

.8 

.1 

380 

4  45 

9  01* 

3  00* 

4.97 

19.92 

.9 

.2 

390 

4  52* 

9  30* 

3  10 

5.38 

21.54 

1.0 

.2 

400 

5  00 

10  00 

3  20 

5.80 

23.23 

1.2 

.2 

410 

5  07* 

10  30* 

3  30 

6.26 

25.00 

1.4 

.2 

420 

5  15 

11  01* 

3  40* 

6.72 

26.86 

1.6 

.3 

430 

5  22J 

11  33* 

3  51 

7.22 

28.82 

1.7 

.3 

440 

5  30 

12  06 

4  02 

7.74 

30.87 

2.0 

.3 

450 

5  37* 

12  39* 

4  13 

8.28 

33.02 

2.2    ' 

.4 

460 

5  45 

13  13* 

4  24* 

8.84 

35.25 

2.4 

.4 

470 

5  52* 

13  48* 

4  36 

9.41 

37.59 

2.7 

.5 

480 

6  00 

14  24 

4  48 

10.03 

40.02 

3.0 

.5 

490 

6  07i 

15  00* 

5  00 

10.67 

42.56 

3.4 

.6 

500 

6  15 

15  37* 

5  12* 

11.33 

45.20 

3.7 

.6 

510 

6  22* 

16  15J 

5  25 

12.03 

47.95 

4.1 

.7 

520 

6  30 

16  54 

5  38 

12.74 

50.79 

4.5 

.8 

530 

6  37* 

17  33* 

5  51 

13.48 

53.76 

5.0 

.8 

540 

6  45 

18  13* 

6  04 

14.26 

56.84 

5.4 

.9 

550 

6  52* 

18  54i 

6  18 

15.07 

60.02 

6.0 

1.0 

560 

7  00 

19  36 

6  32 

15.90 

63  34 

6.5 

1.1 

570 

7  07* 

20  18* 

6  46 

16.76 

66.72 

7.1 

1.2 

580 

7  15 

21  01* 

7  00 

17.65 

70.26 

7.8 

1.3 

590 

7  22* 

21  45* 

7  14* 

18.57 

73.90 

8.4 

1.4 

600 

7  30 

22  30 

7  29 

19.52 

77  68 

9.2 

1  5 

TABLE  VI.— TRANSITION  SPIRAL. 


1°  in  60  ft. 


Length 

D 

A 

8 

0 

y 

x  COR. 

/  COR. 

10 

0°10' 

0°00l' 

0°00' 

.00 

.00 

.0 

0 

20 

0  20 

0  02 

0  00* 

30 

0  30 

0  041 

0  01* 

40 

0  40 

0  08 

0  03 

.03 

50 

0  50 

0  121 

0  04 

.06 

60 

1  0.0 

0  IS 

0  06 

.03 

.10 

70 

1  10 

0  241 

0  08 

.04 

.17 

80 

1  20 

0  32 

0  101 

.06 

.25 

90 

1  30 

0  401 

0  13* 

.09 

.35 

100 

1  40 

0  50 

0  161 

.12 

.48 

110 

1  50 

1  00* 

0  20 

.16 

.64 

120 

2  00 

1  12 

0  24 

.21 

.84 

130 

2  10 

1  24* 

0  28 

.26 

1.06 

140 

2  20 

1  38 

0  321 

.33 

1.33 

150 

2  30 

1  52* 

0  371 

.41 

1.63 

160 

2  40 

2  08 

0  421 

.50 

1.98 

170 

2  50 

2  24* 

0  48 

.59 

2.38 

180 

3  00 

2  42 

0  54 

.70 

2.82 

190 

3  10 

3  00* 

1  00 

.83 

3  32 

200 

3  20 

3  20 

1  061 

.97 

3.88 

210 

3  30 

3  40£ 

1  131 

1.12 

4.48 

.1 

220 

3  40 

4  02 

1  201 

1.29 

5.15 

.1 

230 

3  50 

4  24* 

1  28 

1.47 

5.90 

.1 

240 

4  00 

4  48 

1  36 

1.67 

6.69 

.2 

250 

4  10 

5  12* 

1  44 

1.89 

7.58 

•2 

260 

4  20 

5  38 

1  521 

2.13 

8.52 

.2 

270 

4  30 

6  041 

2  Oil 

2.38 

9.54 

.3 

280 

4  40 

6  32 

2  101 

2.65 

10.64 

.4 

290 

4  50 

7  001 

2  20 

2.94 

11.82 

.4 

300 

5  00 

7  30 

2  30 

3.26 

13.07 

.5 

310 

5  10 

8  00* 

2  40 

3.60 

14.43 

.6 

.1 

320 

5  20 

8  32 

2  501 

3.96 

15.87 

.7 

.1 

330 

5  30 

9  041 

3  Oil 

4.34 

17.40 

.8 

.1 

340 

5  40 

9  38 

3  121 

4.75 

19.02 

.9 

.2 

350 

5  50 

10  12* 

3  24 

5.18 

20.74 

1.1 

.2 

360 

6  00 

10  48 

3  36 

5.64 

22.56 

1.3 

-.2 

370 

6  10 

11  24* 

3  48 

6.12 

24.50 

1.4 

.2 

380 

6  20 

12  02 

4  001 

6.63 

26.53 

1.7 

.3 

390 

6  30 

12  40* 

4  131 

7.16 

28.67 

1.9 

.3 

400 

6  40 

13  20 

4  261 

7  73 

30.92 

2.2 

.4 

410 

6  50 

14  00* 

4  40 

8  34 

33.27 

2.4 

.4 

420 

7  00 

14  42 

4  54 

8.96 

35.73 

2.8 

.5 

430 

7  10 

15  241 

5  08 

9.6i 

38.32 

3.1 

.5 

440 

7  20 

16  08 

5  22J 

10.30 

41.07 

3.5 

6 

450 

7  30 

16  52* 

5  371 

11.01 

43.90 

3.9 

.6 

1°  in  60  ft. 


TABLE  VI.— Continued. 


Length 

D 

J 

0 

0 

y 

x  COR. 

/COR. 

460 

7°40' 

17°38' 

5°52' 

11.75 

46.86 

4.3 

.7 

470 

7  50 

18  24* 

6  08 

12.50 

49.94 

4.8 

.8 

480 

8  00 

19  12 

6  24 

13.35 

53.16 

5  4 

.9 

490 

8  10 

20  00* 

6  40 

14.19 

56.52 

5.9 

1.0 

500 

8  20 

20  50 

6  56 

15.07 

60.01 

6.6 

1.1 

510 

8  30 

21  401 

7  13 

16.00 

63.64 

7.2 

1.2 

520 

8  40 

22  32 

7  30 

16.94 

67.36 

8.0 

1.3 

530 

8  50 

23  24* 

7  471 

17.93 

71.25 

8.8 

1.5 

540 

9  00 

24  18 

8  05 

18.95 

75.31 

9  6 

1.6 

550 

9  10 

25  12i 

8  23 

20.03 

79.53 

10.5 

1.8 

560 

9  20 

26  08 

8  42 

21.13 

83.88 

11.5 

1.9 

570 

9  30 

27  041 

9  00* 

22.26 

88.31 

12.6 

2.1 

580 

9  40 

28  02 

9  19* 

23.42 

92.92 

13.7 

2.3 

590 

9  50 

29  00* 

9  39 

24.67 

97.70 

14.9 

2.5 

600 

10  00     30  00 

9  59 

25.91 

102.66 

16.2 

2.7 

TABLE  VII.     TRANSITION  SPIRAL. 
1°  in  50  ft. 


Length 

D 

J 

e 

o 

y 

x  COR. 

/COR. 

10 

0°12' 

0°00*' 

0°00' 

.00 

.00 

.0 

.0 

20 

0  24 

0  02* 

0  01 

30 

0  36 

0  05i 

0  02 

.02 

40 

0  48 

0  09i 

0  03 

.01 

.04 

50 

1  00 

0  15 

0  05 

.02 

.07 

60 

1  12 

0  211 

0  07 

.03 

.13 

70 

1  24 

0  29* 

0  10 

.05 

.20 

80 

1  36 

0  38* 

0  13 

.07 

.30 

90 

1  48 

0  48*- 

0  16 

.10 

.42 

100 

2  00 

1  00 

0  20 

.15 

.58 

110 

2  12 

1  121 

0  24 

.19 

.77 

120 

2  24 

1  26  J 

0  29 

.25 

1.00 

130 

2  36 

1  411 

0  34 

.32 

1.28 

140 

2  48 

1  57* 

0  39 

.40 

1  60 

150 

3  00 

2  15 

0  45 

.49 

1.96 

160 

3  12 

2  33i 

0  51 

.59 

2.38 

170 

3  24 

2  53* 

0  58 

.71 

2.86 

180 

3  36 

3  141 

1  05 

.85 

3.39 

.1 

190 

3  48 

3  36* 

1  12 

1.00 

3  99 

.1 

200 

4  00 

4  00 

1  20 

1.16 

4  65 

.1 

.0 

1°  in  50  ft. 


TABLE  VII.— Continued. 


a=2. 


Length 

D 

A 

0  . 

0 

y 

;rCOR. 

2fCOR. 

210 

4°  12' 

4°24F 

1°28' 

1.35 

5.38 

.1 

0 

220 

4  24 

4  50i 

1  37 

1.54 

6.19 

.2 

230 

4  36 

5  17* 

1  46 

1.76 

7.07 

.2 

240 

4  48 

5  45i 

1  55 

2.00 

8.04 

'  .2 

250 

5  00 

6  15 

2  05 

2.27 

9.09 

.3 

260 

5  12 

6  45|- 

2  15 

2.55 

10.22 

.4 

270 

5  24 

7  17i 

2  26 

2.85 

11.45 

.4 

280 

5  36 

7  50| 

2  37 

3.18 

12.75 

.5 

290 

5  48 

8  24i 

2  48 

3.54 

14.18 

.6 

.1 

300 

6  00 

9  00 

3  00 

3.91 

15.68 

.7 

.1 

310 

6  12 

9  36-1- 

3  12 

4.32 

17.31 

.9 

.1 

320 

6  24 

10  14-|- 

3  25 

4.75 

19.03 

1.0 

.2 

330 

6  36 

10  53i 

3  38 

5  21 

20.87 

1.2 

.2 

340 

6  48 

11  33i 

3  51 

5.70 

22  81 

1.4 

.2 

350 

7  00 

12  15 

4  05 

6.22 

24.87 

1.6 

.3 

360 

7  12 

12  57i 

4  19 

6.77 

27.05 

1.8 

.3 

370 

7  24 

13  41i 

4  34 

7.34 

29.35 

2.1 

.3 

380 

7  36 

14  26i 

4  49 

7.95 

31.79 

2.4 

.4 

390 

7  48 

15  12* 

5  04 

8.60 

34.35 

2.7 

.4 

400 

8  00 

16  00 

5  20 

9.28 

37.04 

3.1 

.5 

410 

8  12 

16  48£ 

5  36 

10  00 

39.85 

3  5 

.6 

420 

8  24 

17  38i 

5  53 

10.73 

42.79 

4  0 

.7 

430 

8  36 

18  29i 

6  10 

11.53 

45.88 

4  4 

.7 

440 

8  48 

19  211 

6  27 

12.34 

49.14 

5.0 

.8 

450 

9  00 

20  15 

6  45 

13.20 

52.55 

5.6 

.9 

460 

9  12 

21  09J 

7  03 

14.09 

56.05 

6.3 

1.0 

470 

9  24 

22  05i- 

7  21 

15.02 

59.73 

6.9 

1.2 

480 

9  36 

23  02i 

7  40 

15.99 

63.55 

7.7 

.  1.3 

490 

9  48 

24  00* 

8  00 

17.00 

67.55 

8.5 

1  4 

500 

10  00 

25  00 

8  19 

18.05 

71.72 

9.4 

1.6 

510 

10  12 

26  OQi 

8  39 

19.15 

76  00 

10.4 

1.7 

520 

10  24 

27  02| 

9  00 

20.27 

80.04 

11.4 

1.9 

530 

10  36 

28  05i 

9  21 

21.45 

85.08 

12.6 

2.1 

540 

10  48 

29  09* 

9  42 

22.68 

89.88 

13.8 

2.3 

550 

11  00 

30  15 

10  03* 

23.96 

94.85 

15.1 

2.5 

560 

11  12 

31  21* 

10  26 

25.27 

99.97 

16.5 

2.8 

570 

11  24 

32  29* 

10  48 

26.62 

105  19 

18.0 

3.0 

580 

11  36 

33  38J 

11  10* 

28.01 

110.62 

19.6 

3.3 

590 

11  48 

34  48* 

11  34 

29.48 

116.27 

21.3 

3.6 

600 

12  00 

36  00 

11  58 

30.97 

122.13 

23.2 

3  9 

1°  in  40  ft. 


TABLB  Vlli.     TRANSITION  SPIRAL. 


Length 

D 

A 

0 

o 

y 

x  COR. 

t  COR. 

10 

0°15 

0°01' 

0°00' 

"Too" 

"Too"" 

.0 

.0 

20 

0  30 

0  03 

0  01 

30 

0  45 

0  07 

0  02 

.02 

40 

1  00 

0  12 

0  04 

.01 

.05 

50 

1  15 

0  19 

0  06 

.02 

.09 

60 

1  30 

0  27 

0  09 

.04 

.16 

70 

1  45 

0  37 

0  12 

.06 

.25 

80 

2  00 

0  48 

0  16 

.09 

.37 

90 

2  15 

1  01 

0  20 

.13 

.53 

100 

2  30 

1  15 

0  25 

.18 

.73 

110 

2  45 

1  31 

0  30 

.24 

.97 

120 

3  00 

1  48 

0  36 

.31 

1.25 

130 

3  15 

2  07 

0  42 

.40 

1.60 

140 

3  30 

2  27 

0  49 

.50 

2.00 

150 

3  45 

2  49 

0  56 

.61 

2.45 

160 

4  00 

3  12 

1  04 

.74 

2.97 

170 

4  15 

3  37 

1  12 

.89 

3.57 

180 

4  30 

4  03 

1  21 

1.06 

4.24 

.1 

190 

4  45 

4  31 

1  30 

1.25 

4.99 

.1 

200 

5  00 

5  00 

1  40 

1.45 

5,81 

.2 

210 

5  15 

5  31 

1  50 

1  68 

6.72 

2 

220 

5  30 

6  03 

2  01 

1.93 

7.74 

.2 

230 

5  45 

6  37 

2  12 

2.20 

8.85 

.3 

240 

6  00 

7  12 

2  24 

2.51 

10.05 

.4 

250 

6  15 

7  49 

2  36 

2.84 

11.37 

.5 

.1 

260 

6  30 

8  27 

2  49 

3.19 

12.77 

.6 

.1 

270 

6  45 

9  07 

3  02 

3.57 

14.29 

.7 

.1 

280 

7  00 

9  48 

3  16 

3.98 

15.94 

.8 

.1 

290 

7  15 

10  31 

3  30 

4.42 

17-70 

1.0 

.2 

300 

7  30 

11  15 

3  45 

4.89 

19.59 

1.2 

.2 

310 

*7  45 

12  01 

4  00 

5.40 

21.61 

1.4 

2 

320 

8  00 

12  48 

4  16 

5.94 

23.76 

1.6 

.'3 

330 

8  15 

13  37 

4  32 

6.51 

26.05 

1.9 

3 

340 

8  30 

14  27 

4  49 

7.12 

28.46 

2.2 

.4 

350 

8  45 

15  19 

5  06 

7.77 

31.03 

2.5 

.4 

360 

9  00 

16  12 

5  A 

8.46 

33.74 

2.9 

.5 

370 

9  15 

17  07 

5  42 

9.18 

36.62 

3/3 

.5 

380 

9  30 

18  03 

6  01 

9.95 

39.64 

3.7 

.6 

390 

9  45 

19  01 

6  20 

10.75 

42.82 

4.3 

.7 

400 

10  00 

20  00 

6  40 

11.60 

46.16 

4.9 

.8 

410 

10  15 

21  01 

7  00 

12  47 

49.65 

5  5 

.9 

420 

10  30 

22  03 

7  21 

13.39 

53.28 

6.2 

1.0 

430 

10  45 

23  07 

7  42 

14.38 

57.10 

6  9 

1.2 

440 

H  00 

24  12 

8  04 

15.39 

61.12 

7  8 

1.3 

450 

11  15 

25  19 

8  26 

16.45 

65.32 

8.7 

1.5  ( 

1°  in  30  ft. 


TABLE  IX.     TRANSITION  SPIRAL. 


Length 

'  D 

J 

8 

0 

y 

x  COR. 

/COR. 

10 

0°20' 

o°or 

0°00' 

.00 

.00 

.0 

.0 

20 

0  40 

0  04 

0  01 

.00 

.01 

30 

1  00 

0  09 

0  03 

.01 

.03 

40 

1  20 

0  16 

0  05 

.02 

.06 

50 

1  40 

0  25 

0  08 

.03 

.12 

60 

2  00 

0  36 

0  12 

.05 

.21 

70 

2  20 

0  49 

0  16 

.08 

.53 

80 

2  40 

1  04 

0  21 

.12 

.50 

90 

3  00 

1  21 

0  27 

.18 

.71 

100. 

3  20' 

1  40 

0  33 

.24 

.97 

110 

3  40 

2  01 

0  40 

.32 

1.29 

120 

4  00 

2  24 

0  48 

.42 

1.68 

130 

4  20 

2  49 

0  56 

.53 

2.13 

140 

4  40 

3  16 

1  05 

.67 

2.66 

150 

5  00 

3  45 

1  15 

.82 

3.27 

.1 

160 

5  20 

4  16 

1  25 

.99 

3.97 

.1 

170 

5  40 

4  49 

1  36 

1.19 

4.76 

.1 

180 

6  00 

5  24 

1  48 

1.41 

5.65 

.2 

190 

6  20 

6  01 

2  00 

1.66 

6.65 

.2 

200 

6  40 

6  40 

2  13 

1.94 

7.75 

.3 

210 

7  00 

7  21 

2  27 

2.24 

8.97 

.3 

.1 

220 

7  20 

8  04 

2  41 

2.58 

10.31 

.4 

.1 

230 

7  40 

8  49 

2  56 

2.95 

11.77 

.5 

.1 

240 

8  00 

9  36 

3  12 

3.35 

13.38 

.7 

.1 

250 

8  20 

10  25 

3  28 

3.78 

15.11 

.8 

.1 

260 

8  40 

11  16 

3  45 

4.25 

17.00 

1.0 

,2? 

270 

9  00 

12  09 

4  03 

4.76 

19.02 

1.2 

.2 

280 

9  20 

13  04 

4  21 

5.31 

21.20 

1.4 

.2 

290 

9  40 

14  01 

4  40 

5.90 

23.55 

1.7 

.3 

300 

10  00 

15  00 

5  00 

6.53 

26.05 

2.0 

.3 

310 

10  20 

16  01 

5  20 

7.20 

28.72 

2.4 

.4 

320 

10  40 

17  04 

5  41 

7.92 

31.57 

2.8 

.5 

330 

11  00 

18  09 

6  03 

8.69 

34.59 

3.3 

.5 

340 

11  20 

19  16 

6  25 

9.49 

37.80 

3.8 

.6 

350 

11  40 

20  25 

6  48 

10.35 

41.19 

4.4 

.7 

360 

12  00 

21  36 

7  11 

11.25 

44.78 

5.1 

.8 

370 

12  20 

22  49 

7  36 

12.21 

48.56 

5.8 

1.0 

380 

12  40 

24  04 

8  00 

13.22 

52.53 

6.6 

1.1 

390 

13  00 

25  21 

8  26 

14.28 

56.71 

7.6 

1.3 

400 

13  20 

26  40 

8  52 

15.39 

61.10 

8.6 

1.4 

410 

13  40 

28  01 

9  19 

16.56 

65.69 

9.7 

1.6 

420 

14  00 

29  24 

9  47 

17.79 

70.49 

10.9 

1.8 

430 

14  20 

30  49 

10  15 

19.07 

75.51 

12.3 

2.1 

440 

14  40 

32  16 

10  43 

20.41 

80.74 

13.7 

2.3 

450 

15  00 

33  45 

11  13 

21.81 

86.19 

15.4 

2.6 

TABLE  X.     TRANSITION  SPIRAL.     . 


1°  in  20  ft. 


Length 

D 

A 

0 

o 

y 

x  COR. 

/COR. 

10 

0°30' 

0°01' 

0°00' 

.00 

.00 

.0 

.0 

20 

1  00 

0  06 

0  02 

.01 

30 

1  30 

0  13 

0  04 

.01 

.04 

40 

2  00 

0  24 

0  08 

.02 

.09 

50 

2  30 

0  37 

0  12 

.05 

.18 

60 

3  00 

0  54 

0  18 

.08 

.31 

70 

3  30 

1  13 

0  24 

.12 

.50 

80 

4  00 

1  36 

0  32 

.19 

.74 

90 

4  30 

2  01 

0  40 

.26 

1.06 

100 

5  00 

2  30 

0  50 

.36 

1.45 

110 

5  30 

3  01 

1  00 

.48 

1.94 

120 

6  00 

3  36 

1  12 

.62 

2.51 

130 

6  30 

4  13 

1  24 

.79 

3.20 

140 

7  00 

4  54 

1  38 

.99 

3.99 

.1 

150 

7  30 

5  37 

1  52 

1.22 

4.90 

.1 

160 

8  00 

6  24 

2  08 

1.48 

5.96 

.2 

170 

8  30 

7  13 

2  24 

1.78 

7.15 

.3 

180 

9  00 

8  06 

2  42 

2.11 

8.49 

.4 

190 

9  30 

9  01 

3  00 

2.49 

9.98 

.5 

200 

10  00 

10  00 

3  20 

2.90 

11.62 

.6 

.1 

210 

10  30 

11  01 

3  40 

3.36 

13.45 

.8 

.1 

220 

11  00 

12  06 

4  02 

3.86 

15.44 

1.0 

.2 

230 

11  30 

13  13 

4  24 

4.41 

17.63 

1.2 

.2 

240 

12  00 

14  24 

4  48 

5.01 

20.01 

1.5 

.3 

250 

12  30 

15  37 

5  12 

5.66 

22.60 

1.8 

.3 

260 

13  00 

16  54 

5  38 

6.37 

25.38 

2.2 

.4 

270 

13  30 

18  13 

6  04 

7  12 

28  39 

2.7 

.5 

280 

14  00 

19  36 

6  32 

7.94 

31.62 

3.3 

.6 

290 

14  30 

21  02 

•  7  00 

8.82 

35.10 

3.9 

.7 

300 

15  00 

22  30 

7  29 

9.76 

38.83 

4.6 

.8 

310 

15  30 

24  02 

8  00 

10.76 

42.73 

5.4 

.9 

320 

16  00 

25  36 

8  31 

11.82 

46.92 

6.3 

1.1 

330 

16  30 

27  13 

9  04 

12.95 

51.36 

7  4 

1.2 

340 

17  00 

28  54 

9  37 

14.15 

56.05 

8.6 

1.4 

350 

17  30 

30  37 

10  11 

15.43 

61.09 

9.9 

1.7 

360 

18  00 

32  24 

10  46 

16.75 

66.31 

11.3 

1.9 

370 

18  30 

34  14 

11  19 

18.16 

71.63 

13.0 

2.2 

380 

19  00 

36  06 

12  00 

19.65 

77.35 

14.8 

2.5 

390 

19  30 

38  02 

12  38 

21.21 

83.41 

16  8 

2.8 

400 

20  00 

40  00 

13  17 

22.87 

89.83 

19.0 

3.2 

TABLE  XI.     TRANSITION  SPIRAL. 


1°  in  10  ft. 


Length 

D 

A 

B 

0 

y 

^COR. 

/COR 

10 

1°00' 

0°03' 

0°01' 

.00 

.00 

.0 

.0 

20 

2 

0  12 

0  04 

.01 

.02 

30 

3 

0  27 

0  09 

.02 

.08 

40 

4 

0  48 

0  16 

.05 

.19 

50 

5 

1  15 

0  25 

.09 

.36 

60 

6  00 

1  48 

0  36 

.16 

.63 

70 

7 

2  27 

0  49 

.25 

1.00 

80 

8 

3  12 

1  04 

.37 

1.49 

90 

9 

4  03 

1  21 

.53 

2.12 

100 

10 

5  00 

1  40 

.73 

2.91 

.  1 

110 

11  00 

6  03 

2  01 

.97 

3  87 

.1 

120 

12 

7  12 

2  24 

1.26 

5.02 

.2 

130 

13 

8  27 

2  49 

1.60 

6.38 

.3 

140 

14 

9  48 

3  16 

1.99 

7.97 

.4 

.1 

150 

15 

11  15 

3  45 

2.45 

9.79 

.6 

.1 

160 

16  00 

12  48 

4  16 

2.97 

11.87 

.8 

.1 

170 

17 

14  27 

4  49 

3.56 

14.23 

1.1 

.2 

180 

18 

16  12 

5  24 

4.23 

16.87 

1.4 

.2 

190 

19 

18  03 

6  01 

4  97 

19.81 

1  9 

.3 

200 

20 

20  00 

6  39 

5.79 

23.07 

2.4 

.4 

210 

21  00 

22  03 

7  20 

6.70 

26.65 

3.1 

.5 

220 

22 

24  12 

8  03 

7.69 

30.58 

3.9 

.6 

230 

23 

26  27 

8  48 

8.78 

34.86 

4.8 

.8 

240 

24 

28  48 

9  35 

9.96 

39.49 

6  0 

1.0 

250 

25 

31  15 

10  23 

11.24 

44.49 

7.3 

1.2 

TABLE  XII.  FACTORS  FOR  ORDINATBS. 

To  find  jv,  multiply  o  by  the  factor  for  the  ratio  found  by 
dividing  the  distance  of  the  point  from  the  P.  S.  or  P.  C.  C. 
by  the  half-length  of  spiral. 


Ratio  to 
Yz  length 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1  C' 

Factor... 

.0005 

.004 

.014 

032 

063 

108 

172 

.256 

.365 

.500 

TABLE  XIII. 

SPIRAL  DEFLECTION  ANGLES    FOR  FIRST 
CHORD  LENGTH. 

Values  of  6l  in  minutes  for  use  with  Table  XIV. 

The  first  line  of  captions  gives  the  distance  in  feet  in  which  1°  of  degree 
of-curve  is  attained.  The  second  line  of  captions  gives  the  value  of  a« 
Angles  are  given  in  minutes 


Chord 

200 

150 

125 

100 

SO 

75 

Chord 

Length 

X 

2A 

%  - 

1 

IX 

.    IK 

Length 

10 

.050 

.067 

.080 

.100 

.125 

.133 

10 

11 

.060 

.081 

.097 

.121 

.151 

.161 

11 

12 

.072 

.096 

.115 

.144 

.180 

.192 

12 

13 

.084 

.113 

.135 

.169 

211 

.225 

13 

14 

.088 

.131 

.157 

196 

.245 

.261 

14 

15 

.112 

.150 

.180 

.225 

.281 

.300 

15 

16 

.128 

.171 

.205 

.256 

.320 

.341 

16 

17 

.144 

.193 

.231 

.289 

.361 

385 

17 

18 

.162 

.216 

.259 

.324 

.405 

.432 

18 

19 

.180 

.241 

.289 

.361 

.451 

.481 

19 

20 

.200 

.267 

.320 

.400 

.500 

533 

20 

25 

.312 

.417 

.500 

.625 

.781 

.833 

25 

30 

.450 

.600 

.720 

.900 

1  125 

1.200 

30 

35 

.612 

.817 

.980 

1.225 

1.531 

1.633 

35 

40 

.800 

1.067 

1.280 

1.600 

2.000 

2.133 

40 

45 

1.012 

1.350 

1.620 

2.025 

2.531 

2.700 

45 

50 

1.250 

1.667 

2.000 

2.500 

3.125 

3.333 

50 

Ch<rrd 

60 

50 

40 

30 

25 

20 

Chord 

Length 

1% 

2 

2/2 

3K 

4 

5 

Length 

10 

.167 

.200 

.250 

.333 

.400 

.500 

10 

11 

.202 

.242 

.302 

.403 

.484 

'.605 

11 

12 

.240 

.288 

.360 

.480 

.576 

.720 

12 

13 

.282 

.338 

.422 

.563 

.676 

.845 

13 

14 

.327 

.392 

.490 

.653 

.784 

.980 

14 

15 

.375 

.450 

.562 

.750 

.900 

1.125 

15 

16 

.427 

.512 

.640 

.853 

1.024 

1.280 

16 

17 

.482 

.578 

.722 

.963 

1.156 

1.445 

17 

18 

.540 

.648 

.810 

1.080 

1.296 

1.620 

18 

19 

.602 

.722 

.902 

1.203 

1.444 

1.805 

19 

20 

.667 

.800 

1.000 

1  333 

1.600 

2.000 

20 

25 

1.042 

1.250 

1.562 

2.083 

2.500 

3.125 

25 

30 

1.500 

1.800 

2.250 

3.000 

3.600 

4.500 

30 

35 

2.042 

2.450 

3.062 

4.083 

4  900 

6.125 

35 

40 

2.667 

3.200 

4.000 

5.333 

6.400 

8.000 

40 

45 

3.375 

4.050 

5.062 

6.750 

8.100 

10.125 

45 

50 

4.667 

5.000 

6.250 

8.333 

10.000 

12.500 

50 

TABLE  XIV,  COEFFICIENTS  FOR  DEFLECTION  ANGLES. 

To  find  the.  deflection  angle  from  tangent  at  the  instrument  point,  multiply  the  spiral  deflection  angle  at  the 
±*.  S.  for  a  single  chord  length  by  the  coefficient  found  in  the  instrument  point  column  opposite  the  number  of  the 
chord  point  to  be  located. 

wTEJw 

o~«««»o 

cot-ooo>o 

rH 

rHrHrHrHpH 

INSTRUMENT  AT  CHORD  POINT  NUMBER 

10 
r—  I 

iO  co  rH  ON  I^N.  10 
ft  ft  ft  CO  CO  CO 

ft  VO  VO  ft  O 
cxi  ON  VO  co  O 
CO  CM  CM  CM  CM 

ft  vO  VO  ft 

vo  CM  oo  ftr-s 

rH  rH                 V* 

rH 

CM  t^  O  rH  O  1^ 

ON  t^  VO  ft  Cxi  ON 
CO  co  co  co  co  CM 

CM  iO  VO  tO  CM 

t^   ft   rH   CO   tO 
CM  CM  CM  rH  rH 

gSSoS? 

CO 
rH 

CO  ft  CO  O  O  CO 
co  CM  O  ON  !>.  ft 
CO  CO  CO  CM  CM  CM 

ft  CO  O  O  CO 
CM  ON  t^  ft  O 

CM  -H   rH  r-i    rH 

SS0?S3 

rH 

OO  iO  O  co  ft  <O 

co  r^  vo  ft  CM  o 

CM  CM  CM  CM  CM  CM 

O  >O  CO  ON  00 
CO  to  CM  ON  VO 
rH  rH  rH 

CO  /—  s  co  t>.  i—  ' 
Q-p                rH 

rH 
rH 

CM  O  VO  O  CM  CM 
ft  CO  rH  O  CO  VO 
CM  CM  CM  CM  rH  rH 

O  VO  O  CM  CM 
ft  rH  ON  VO  CO 
rH  rH 

ft  O  00  CO 
Q  co  1>.  O  ft 

O 

iH 

§ON  VO  <H  ft  tO 
CO  t^  VO  ^T  CM 
CM  rH  rH  rH  rH  rH 

ft  rH  VO  ON 

o  oo  to  <MO 

iH 

•H  ft  ON  VO  to 
f«)  VO  ON  co  l^ 

rH  rH 

0 

CM  CM  O  VO  O  CM 
VO  to  ft  CM  rH  ON 

CM  O  VO         00 
t*-  to  CMQ  CM 

CO  O  ft  O  CO 
iO  ON  CM  v£?  ON 
rH  rH  rH 

OO 

00  ON  00  iO  O  co 
CM  rH  O  ON  CO  VO 
rH  rH  rH 

ft  CO         10  CM 

«-H  CM  iO  O  l^ 
00  rH  ft  CO  rH 
rH  rH  iH  CM 

- 

00  O  O  CO  "H-  00 
ON  ON  CO  vO  iO  CO 

O         CM  VO  CM 
CMOCM  ft  ^ 

8O  CM  VO  CM 
CO  VO  ON  co 
rH  rH  rH  rH  CM 

CO 

CM  to  VO  tO  CM  £>• 

ON  O  co  00 

IO  -H-  10  CO  CO 
rH  iH  rH  CM  CM 

LO 

O  ft  VO  VO  ft 
IO  T  CO  CM  »HQ 

VC  ft  ft  VO  O 
rH  CO  to  J>»  O 
rH 

VO  ft  ft  VO  O 
CM  IO  GO  rH  IO 
rH  rH  rH  CM  CM 

ftf 

CM  t^  O  rH         CO 
CO  CM  CM  rH  O1"""1 

OO  IO  ft  IO  00 
CM  ft  VO  OO  O 

CO  O  ON  O  co 
CO  VO  CO  CM  iO 
rH  rH  rH  CM  CM 

OO 

OO  ft  CO         O  CM 
i-H  rH          Q  rH  CM 

VO  CM  O  O  CM 
CO  IO  t^  ON  rH 
rH 

VO  CM  O  O  M 

CO  VO  ON  CM  to 

iH   rH   rH   CM   CM 

oq 

O          """^ 

O  10  CM  rH  CM 
ft  10  t^-ONrH 

to  o  t>i  vo  t^ 

CO  VO  CO  rH  ft 
rH  rH  rH  CM  CM 

rH 

CM         ft  O  00  00 

O           rH   rHCM 

O  ft  O  00  00 
ft  IO  t^  OO  O 
rH 

O  ft  O  CO  CO 
CO  iO  CO  O  CO 
rH  rH   r-l  CM  CM 

O 

rH  ft  ON  VO  »O 

o             IHCSI 

VO  ON  ft  rH  o 
CO  ft  VO  00  O 
rH 

rH  ft  ON  VO  IO 
CM  ft  VO  ON  CM 
rH  rH  rH  rH  CM 

pioj  pjoq<) 

OrHOqOXtflO 

COt-CQOSO 
rH 

rH  rH  rH  rH  rH 

TABLE  XV.     STREET  RAILWAY  SPIRAL. 
£=2000.  £=2000. 


Length 

Radios 

A 

0 

0 

y 

.rCoR. 

/COR. 

5 

400 

0°21' 

0°07' 

.01 

.00 

.00 

10 

200 

1  26 

0  29 

.08 

.00 

15 

133.33 

3  13 

1  04 

.28 

.00 

16 

125 

3  40 

1  13 

.34 

.01 

20 

100 

5  44 

1  55 

.17 

.67 

.02 

21.06 

95 

6  21 

2  07 

.19 

.78 

.03 

22.22 

90 

7  04 

2  21 

.23 

.91 

.03 

.00 

23.53 

85 

7  56 

2  39 

.27 

1.09 

.04 

-.01 

25 

80 

8  57 

2  59 

.32 

1.30 

.06 

.01 

26.67 

75 

10  11 

3  24 

.39 

1.58 

.08 

.01 

28.57 

70 

11  41 

3  54 

.48 

1.94 

.12 

.02 

30 

66.67 

12  54 

4  18 

.56 

2.24 

.15 

.02 

30.77 

65 

13  34 

4  31 

.61 

2.42 

.17 

.03 

33  33 

60 

15  55 

5  18 

.77 

3.07 

.26 

.04 

36.36 

55 

18  56 

6  19 

1.00 

3.98 

.40 

.07 

40 

50 

22  55 

7  38 

1.32 

5  27 

.64 

.11 

44.44 

45 

28  17 

9  25 

1.81 

7.19 

1.08 

.18 

50 

40 

35  49 

11  54 

2.56 

10.13 

1.95 

.32 

55 

36.36 

43  20 

14  23 

3.39 

13.29 

3.15 

.52 

56.05 

35.68 

45  00 

14  55 

3.59 

14-02 

3  46 

.58 

TABLE  XVI. 

£=1500. 


STREET  RAILWAY  SPIRAL. 


=  1500. 


Length 

Radius 

A 

0 

0 

y 

.r  COR. 

/COR. 

5 

300 

0°29' 

0°10' 

.01 

.00 

.00 

10 

150 

1  55 

0  38 

.11 

.00 

12 

125 

2  45 

0  55 

.19 

.CO 

t  f 
±\j 

100 

4  18 

1  26 

.38 

.01 

15.79 

95 

4  46 

1  35 

.43 

.01 

16.67 

90 

5  19 

1  46 

.51 

.01 

17.65 

85 

5  58 

1  59 

.61 

.02 

18.75 

80 

6  43 

2  14 

.18 

.73 

.03 

20 

75 

7  38 

2  33 

.22 

.89 

.03 

.00 

21.43 

70 

8  46 

2  55 

.27 

1.09 

.05 

.01 

23.08 

65 

10  10 

3  23 

.34 

1.36 

.07 

.01 

25 

60 

11  56 

3  59 

.43 

1.74 

.11 

.02 

27.27 

55 

14  13 

4  44 

.56 

2.24 

.17 

.03 

30 

50 

17  11 

5  44 

.75 

2.98 

.27 

.04 

33.33 

45 

21  13 

"7  04 

1.02 

4.07 

.45 

.07 

35 

42.86 

23  24 

7  47 

1.18 

4.71 

.57 

.09 

37.50 

40 

26  51 

8  56 

1.45 

5.77 

.82 

.14 

40      • 

37.50 

30  34 

10  10 

1.76 

6.96 

1.14 

.19 

42.86 

35 

35  05 

11  40 

2.16 

8.51 

1.61 

.27 

45 

33.33 

38  41 

12  51 

2.49 

9.80 

2.05 

.34    i 

TABLE  XVII.     STREET  RAILWAY  SPIRAL. 
£-^1250  £  =  1250 


Length 

Radius 

4 

0 

0 

y 

x  COR. 

/COR. 

5 

250 

0°34' 

o°ii' 

.02 

.00 

.00 

10 

125 

2  17 

0  46 

.13 

15 

83.33 

5  11 

1  44 

.45 

.01 

15.62 

80 

5  36 

1  52 

.51 

.01 

16.67 

75 

6  22 

2  07 

.15 

.62 

.02 

17.86 

70 

7  19 

2  26 

.19 

.76 

.03 

.00 

19.23 

65 

8  29 

2  50 

.24 

.95 

.04 

.01 

20 

62,50 

9  10 

3  03 

.26 

1.04 

.05 

.01 

20.83 

60 

9  56 

3  19 

.30 

1.20 

.06 

.01 

22.73 

55 

11  50 

3  57 

.39 

1.56 

.10 

.02 

25 

50 

14  19 

4  46 

.52 

2  07 

.15 

.02 

27.78 

45 

17  41 

5  54 

.71 

2.84 

26 

.04 

30 

41  66 

20  38 

6  52 

.89 

3.56 

38 

.06 

31.25 

40 

22  24 

7  28 

1.01 

5.04 

.47 

.08 

35 

35.71 

28  05 

9  20 

1.40 

5.62 

.83 

.14 

TABLE  XVIII. 
-1000 


STREET  RAILWAY  SPIRAL. 

£  =  1000 


Length 

Radius 

A 

8 

o 

y 

x  COR. 

/COR. 

5 

200 

0°43' 

0°14' 

.02 

.00 

.00 

10 

100 

2  52 

0  57 

.17 

.01 

15 

66.67 

6  27 

2  09 

.56 

.02 

15.39 

65 

6  47 

2  16 

.15 

.61 

.02 

16.67 

60 

7  57 

2  39 

.19 

.77 

.03 

.00 

58.18 

55 

9  28 

3  09 

.25 

1.00 

.05 

.01 

20 

50 

11  27 

3  49 

.33 

1  33 

.08 

.01 

22.22 

45 

14  09 

4  44 

.46 

1.83 

.13 

.02 

25 

40 

17  54 

5  57 

.64 

2.58 

.24 

.04 

28.57 

35 

23  23 

7  47 

.97 

3.84 

.47 

.08 

30 

33.33 

25  47 

8  34 

1.12 

4.45 

.60 

.10 

33.33 

30 

31  50 

10  35 

1.53 

6.05 

1.03 

.17 

TABLE  XIX.     STREET  RAILWAY  SPIRAL, 

£  =  750  £  =  750 


Length 

Radius 

A 

8 

0 

y 

x  COR. 

/COR. 

5 

150 

0°57' 

0°19' 

.03 

.00 

.00 

i  r\ 
±\J 

75 

3  49 

1  16 

.22 

.01 

15 

50 

8  54 

2  52 

.19 

.75 

.03 

.00 

16.67 

45 

10  37 

3  32 

.26 

1.03 

.06 

.01 

18.75 

40 

13  24 

4  28 

.36 

1.46 

.10 

.02 

20 

37.5 

15  17 

5  05 

.44 

1.75 

.14 

.02 

21.43 

35 

17  32 

5  50 

.57 

2.27 

.20 

.03 

25 

30 

23  52 

7  -S7 

.86 

3.43 

.43 

.07 

TABLE  XX.     OFFSETS  FOR  SPIRALS. 


I) 

3 

4° 

5° 

6° 

7° 

a 

0 

L 

0 

L 

0 

L 

0 

L 

0 

L 

0.5 

7  83 

6.000 

0.6 

5  44 

5.000 

12.86 

6.667 

0.7 

4  00 

4  286 

9.47 

5.714 

18.47 

7-143 

0.8 

3.06 

3-750 

7-24 

5-000 

14-06 

6.250 

24-34 

7.500 

38-46 

8.750 

0.9 

2-41 

3-333 

5.71 

4-444 

11.14 

5-555 

19-22 

6-667 

30.41 

7.778 

1.0 

1.96 

3.000 

4.64 

4-000 

9.04 

5-000 

15-60 

6.000 

24.71 

7.  000 

1.1 

1.63 

2.727 

3.84 

3-636 

7.48 

4.546 

12.88 

5.455 

20-42 

6-364 

12 

1.36 

2.500 

b.c2 

3-333 

6.30 

4.167 

10.84 

5-000 

17-16 

5-833 

1.3 

1.16 

2.308 

2.75 

3-077 

5-36 

3  846 

9.26 

4.615 

14-63 

5-384 

1.4 

1  00 

2.143 

2.37 

2-857 

4.61 

3-572 

7.99 

4-286 

12  65 

5-000 

1.5 

.87 

2.000 

2.07 

2.667 

4  01 

3  334 

6.95 

4.000 

11-04 

4.667 

1.6 

1.81 

2.500 

3-51 

3.125 

6.11 

3-750 

9-71 

4.375 

1.7 

1.60 

2  353 

3.13 

2.941 

5-41 

3-529 

8-59 

4.117 

1.8 

1.43 

2.222 

2-80 

2.778 

4.81 

3-334 

7-62 

3-889 

1.9 

1.30 

2.105 

2.50 

2.632 

4.35 

3.158 

6-90 

3-684 

2.0 

1.16 

2.000 

2.27 

2-500 

3.91 

3.000 

6.22 

3.500 

2.2 

.96 

1.818 

1.88 

2.272 

3.22 

2-727 

5-13 

3.182 

2-4 

.80 

1.667 

1.57 

2.084 

2-72 

2-500 

4-33 

2.917 

2.6 

1.20 

1.923 

2.32 

2-308 

3.67 

2.692 

2.8 

1.14 

1-785 

1.99 

2.143 

3.17 

2.500 

3.0 

1.00 

1.667 

1.74 

2.000 

2.78 

2.333 

D 

8° 

9° 

1O° 

11 

12° 

a 

o 

L 

0 

L 

o 

L 

0 

L 

o 

L 

1.0 

36.70 

8-000 

1.2 

25-53 

6.667 

1.4 

18.80 

5-714 

26.72 

6.429 

1.6 

14.45 

5-000 

20.47 

5-625 

1-8 

11.40 

4.445 

16.23 

5-000 

22.15 

5.556 

2.0 

9.28 

4.000 

13  20 

4.500 

18.05 

5.000 

2-2 

7.67 

3.636 

10.90 

4.090 

14.94 

4  545 

19-78 

5.000 

2.4 

6.43 

3.337 

9-17 

3-750 

12.50 

4-167 

16.63 

4.584 

21  54 

5.000 

2.6 

5.47 

3.077 

7-81 

3.461 

10.64 

3-846 

14-19 

4.231 

18  43 

4.615 

2.8 

4.74 

2.857 

6.77 

3.214 

9.21 

3-571 

12.19 

3.927 

15.83 

4.  285 

3-0 

4.14 

2.667 

5.88 

3.000 

8-02 

3.333 

10.70 

3.667 

13-84 

4  000 

3-2 

3.62 

2-500 

5.15 

2.812 

7-04 

3-125 

9.39 

3.438 

12.15 

3-750 

3-4 

3.20 

2.353 

4-58 

2-647 

6.25 

2-941 

8.34 

3  234 

10.79 

3-528 

3-6 

2.86 

2.222 

4.08 

2  500 

5.59 

2-778 

7.44 

3.056 

9.61 

3-334 

3-8 

2.56 

2-106 

3.67 

2.369 

5-01 

2.632 

6.64 

2.895 

8-67 

3.158 

4.0 

2.32 

2.000 

3-30 

2.250 

4-52 

2.500 

6.02 

2.750 

•7.81 

3  000 

4.5 

1.84 

1.778 

2.61 

2.000 

3-57 

2-222 

4.74 

2.444 

6.20 

2.667 

5.0 

1.48 

1.600 

2  11 

1-800 

2.90 

2.000 

3.86 

2.200 

5.01 

2.400 

5-5 

1.23 

1.454 

1-75 

1  636 

2.40 

1-818 

3  19 

2.000 

4.14 

2.182 

6-0 

1.03 

1.3J4 

1.47 

1.500 

2  02 

1  667 

2.63 

1.834 

3.48 

2.000 

X 


>ATE 


LIBRARY  USE 

TURN  TO  DESK  FROM  WHICH  BORROWED  f  TS 

LOAN  DEPT. 

HIS  BOOK  IS  DUE  BEFORE  CLOSING  TIME 


ON  LAST  DATE  STAMPED  BELOW 


TURN 

ALTY 
JRTH 
DAY 


JN25  1972  0 


LD62A-30m-2,'71 
(P2003slO)9412A-A-32 


General  Library 

University  of  California 

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Y  A 


380514 


OF  CALIFORNIA  LIBRARY 


